T N the establishment of a strong position the Ps are perhaps the most important of all the pieces. They serve at first as the fortifications in whose embrasures and behind whose sheltering walls the other pieces are deployed; and later they are brought into use as active agents and formidable participants in the contest. Naturally, therefore, in the theory of minor tactics, the Ps are first to be considered ; and, with the aid of the board and pieces, which the learner is again urged most assiduously to employ in his study, we will now proceed to the examination of the fundamental positions of Ps. If you will place the white Ps on the board in their normal position, (cf. p. 45), leaving off all the other pieces, and will advance the QBP one step, (Q B P- Q B 3), and Q P two steps (Q P- Q4), you will have created the first distinctive P position; Q Kt P, Q B P, and Q P here occupy three consecutive points upon a diagonal, extending outward from the remaining Ps, which, being unmoved, occupy points in a straight line. To the latter Ps, taken collectively, we apply the term normal Pbase; to the former three, the term salient. It will be noticed that the salient and the normal P base have one P (Q Kt P) in common. If you will next advance KP two steps (KP-K4), you will have another characteristic formation, consisting of QP and KP, which we designate a supporting parallel; and, considering the salient and the supporting parallel together, we bestow upon this formation the title angle of resistance. The technical terms that we have introduced will be of great convenience in the construction of our theory, and we shall formally define them as follows: —

A salient is a formation of Ps occupying three consecutive points upon a diagonal, the initial point of which is in the second horizontal of the board.

A supporting parallel is a formation of Ps occupying two consecutive points upon the fourth horizontal.

An angle of resistance is a formation of Ps consisting of the combination of a salient and a supporting parallel which have a P in common.

The normal P base consists of those Ps which are left in their normal positions after the formation of a salient, a supporting parallel or an angle of resistance.

If one of these elementary P formations is more valuable than the others, it is the salient, which is the prime requisite of a sound defensive position, or — *ince the base of a true line of attack always constitutes the base of a true line of defence, — of a sound offensive position. In the salient that we have just established, QBPat QB3 repels any radius of offence that may be operated against K, before castling, along the K's major diagonal, by the adverse K B and Q; and QP at Q4 similarly repels any radius of offence that may be operated against K B 2 and KKt 1, or against K itself, after castling, along the K Kt's major diagonal, by the same adverse pieces. Add to this salient KP at K4, forming with QP at Q4 a supporting parallel, and completing the angle of resistance ; and no great experience in chess play is necessary to convince the student, not only that this means of sheltering K from the adversary's assaults is safe and sufficient, but also that the defensive position thus constructed is well-nigh impregnable. This formation may be still farther strengthened, however, for purely offensive measures, by the advance of KBP two steps (KBP-KB4) ; in but few circumstances should this be done before castling, inasmuch as K in its normal position, if KBP has been moved, is exposed to the operation of a radius of offence along the K's minor diagonal. Such an attack may be repelled, to be sure, by the advance of KKtP one step (KKt P -KKt3), or it may be prevented by the posting of K Kt at K B 3 ; nevertheless, there are difficulties attending cither of these methods of defence, which need not be dwelt upon here : it is enough to remember that the functions of Ps in minor tactics are essentially defensive, and that a sally like KBP -KB4, although a powerful manœuvre for offence, is theoretically defective if essayed before castling, on the ground that it exposes K, instead of sheltering it. Therefore, as a rule, whenever the advance of KBP, whether one or two steps, is recommended in this theory, the previous castling of K is to be taken for granted. When KBP has been advanced two steps, it forms, in conjunction with K P at K4 and QP at Q4, what may be called a co7nposite supporting parallel, and the angle of resistance consisting of this supporting parallel and the salient that has been described, is called a composite angle of resistance. The composite angle of resistance is the only elementary P formation in which three Ps are posted abreast on the fourth horizontal of the board, and it is by far the most powerful of all the P formations.

The next salient to which the consideration of the student is invited consists of K Kt P in its normal position, KBP advanced one step (KB P-KB 3), and KP advanced two steps (KP-K4). Between this salient and that which has previously been described, an intimate relation exists : the salient consisting of K Kt P, K B P and K P should be established only after the establishment of the salient consisting of Q Kt P, Q B P and Q P ; for which reason the former is denominated an auxiliary salient. The formation of both these salients, it will be noticed, presents two angles of resistance, which have in common the supporting parallel consisting of Q P at Q4 and KP at K4. Together with the unmoved Ps that constitute the normal P base, this position of Ps possesses a character of completeness that entitles it to the specific designation of a P integral ; in contradistinction to which term the elementary formations that enter into its construction are called P units. 1 he following diagram (Fig. 24) and formula clearly show the composition of this P integral, in which we include K in order to show that it should occupy its position after castling (K R) : —

Formula: P integral = P unit (element of normal P base-Q R P, Q Kt P) plus P unit (salient-Q Kt P, Q B P, Q P) plus P unit (supporting parallel-Q P, K P) plus P unit (salient-K P, KBP, K Kt P) plus P unit (element of normal P base-K Kt P, KRP).

We will now consider the formation of all the P units and P integrals in detail. The necessity of their diligent examination, one by one, cannot be too strenuously urged upon the student of minor tactics.

The salients are six in number, and are

A Pawn Integral.

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