Dec. 15, 1882. 



KNO^A^LEDGE 



473 



&UV iJflattjtmattcal Column* 



EAST LESSOXS IN THE DIFFEEENTIAL CALCULUS. 

 Xo. XV. 



WE have next to consider the various methods available for 

 integrating the different expressions which come before us 

 when we apply the methods of the integral calculus. 



First some expressions can be integrated at once, because they 

 have been already obtained as the differential coefficients of known 

 functions. For these the following table, which gives aU the 

 differential coefficients of simple functions, will be found useful, 

 and should always be held in the student's remembrance. We write 



— (function of x) for the differential coefficient of the function 



with respect to i. 



Differential Coefficient. 



dx 



|-(Iog^)=i; 

 ax X 



i.Oog^) = _ 



^(a') = a'.log^; 



— (sin a:) = cos x; 

 dx 



£(coa»)--sinz 



±{cotx)=Jl± 

 da Bui'x 



l(sec«)=!i^^ 

 dx cos'a: 



-i(cosecx) = Zi 

 dx SI 



= tan X 

 ' — cot a? 



Deduced Integi-al. 



^_=log^ 



J— = ^°S<^ ■ ]og,x = I<^^ 



Jl'dx^e' 



J log,a 



/cos X diC = sin j- 



/sin a; d jc = — cos ■■ 

 J cos-x 



/-^ 



J sm-j 



/li^£ff=secx 

 J cos'x 



/cos i' ''2: _ _ 

 •J sin'jt 

 ACBin-.^)=_^ fJl^^^^. 



lCtan-£)=_^; /li^ = ltan-f 

 dA a/ a' + j=' Jd}-^x- a a 



A(cOt-'?) =-r£. ; rjf^ = -Icot-'f 



da\ aJ a' + x' Ja' + x'' a a 



d( _,\ —ardx 1 X 



The last six relations are, of course, reducible to three only, so 

 far aa integration is concerned. Since 



sin"*- + cos~- = tan"'- + cot"' - = sec"- - + cosec~'-= — , 

 a a a a a a % 



it is manifest that the second values given respectively to 

 /-^^ fj^ and / ^_ 

 ■y ^a'-x' J a^-^ii' J xv/x'-a»' 



differ from the first only by a constant,- 



three cases, and — in the other three cases. 



Bvx Cfifsfs Column. 



By Mepoisto. 



REPRINTS. 



THE following position, sent us by Mr. Pearson, is (probably) the 

 original of Eichstadt's problem. The ix>sition is as follows :— 

 White.— King on QR7, Rook on QKtS, Knights on Q5 and K7. 

 Black.— King on QR4, Knight on QK7. 



White to play and mate in three moves. 



The following problem by the late Eov. H. Bolton was sent to us 

 by Mr. Miles :— 



White.— King on QB8, Book on KB5, Bishop on KKt3, Pa?rns on 

 KKt2, QB4, QR3. 



Black.— King on QB3, Pawns on QB5, KKto. 



White to play and mate in four moves. 



The following game was played by Mr. SteLnitr, who is now on 

 a visit to Philadelphia : — 



steinitz gambit. 



White. Black. 



Steinitz. Capt. Micbaelis. 



1. P to K4 P to K-1 



2. QKt to B3 QKt to B3 (a) 



3. P to B4 P takes P 



4. P to Q4 (6) Q to R5 (cli) 



5. K to K2 P to QKt3 (c) 

 0. Kt to Kt5 B to R3 



7. P to B4 B takes Kt 



8. P takes B Q to R4 (ch) 



9. Kt to B3 Q takes P (ch) 



0. K to B2 Q to Kt5 



1. P to QR3 Q to K2 



2. B takes P Q takes P 



Wiite. Black. 



Sleinitz. Capt. Miihaelia. 



1 13. Q to Q2 (d) Castles (e) 

 I 14. B to BC (ch) K to Kt sq. 



15. KRtoKsq. Q to Q4 

 ' IG. QR to B sq. Kt to B3 

 ; 17. Q to B2 (/) B to Q3 



18. B to QB4 Kt takes P (j) 



19. Q to Q3 Q to KB4 (h) 



20. B takes B Q takes Q 



21. B takes Q Kt takes Kt 



22. BtksP(ch(i)K toKt2 



23. P takes Kt R to QB sq. 



24. B to K5 and wins. 



NOTES. 



(a) We prefer KKt to B3. Also B to B4 leads to an even game. 



(6) This constitutes the Steinitz gambit. We do not consider it 

 sound play. White thinks to be able to develop by Kt to B3, and 

 then bringing his King to B2. 



((■) Black must play carefully, as otherwise White will speedily 

 bring his pieces into play, besides having two strong centre 

 Pawns. White also threatens to win the weak KBP. Black's best 

 move at this juncttu-e is 5. P to Q4, which White might follow up 

 by either 6. Kt takes P or 6. P takes P. If 5, 



^ Kt takes P 



Kt to B3 



B takes P 

 "K^ 



P to Q4 

 — , and Black has the 



B to Kt5 (ch) Castles 

 advantage, for he will win one of the centre Pawns. 9. Kt takes 

 Kt would be no good, on account of Kt takes P (ch) ; if P to B3 or 

 B4, then KKt takes P, or if B to Kt3, then Q to B3. Also B takes 

 T,r> ■ r 1 ■ n B takes P in P takes R 



BP IS of no avail, i.e., 9. — ^ 10. -— ; - 



R takes Kt KKt takes P 



11. ° ^^ ^° to prevent Kt takes P (ch) or B takes Kt (ch) 



11. 



P takes Kt 



„ , „ „ (as the White Qneen cannot take 



Kt takes B B to QB5 *■ 



the Knight, on account of Q to B7 (ch), followed by R to Q sq. 



Q to K sq. , . K to Q sq. 



• RtoQsq.(ch) 



winning the Queen) 



15. 



toQ3 



16. 



KttoB 

 Q takes Q 



(ch) 

 17. 



R to B sq . 



Kt takes KtP Kt takes Q ' Kt takes Kt and wins. 



If again, in reply to the proposed move of 5. P to Q4, Whit e shou ld 

 plav 6. P takes P, the game would proceed thus : — 5. P to Q4> 

 P takes P „ Kt to B3 g P_to^Q6^ g B takes P 



Kt to K2 ' P takes P ' Kt i 



Kl5 (ch) 



7. 



B4 



10. 



K toQ3 



, and we think Black has no disadvantage. 



Q to R5 



{d) Threatening £ to K sq., B takes P would not be good, on 

 account of P to Q3. 



(e) White has a great facility for deploying his pieces ; for that 

 reason Steinitz believes in this risky gambit. Castles has the 

 disadvantage of exposing the King to the attack of the two Bishops, 

 but, as in most of Steinitz's games, although apparently ho 

 threatens nothing in particular, yet it is difficult to see what to do. 

 If Q to Q4, then White would obtain a good game by R to B sq., 

 threatening by B to B4, R to K sq., or P to Q5, to bring all his 

 pieces into tino play, and make things in general uncomfortable for 

 Black. Our choice would have been 13. Kt to B3, if now 14. R to 

 K s([., then B to K2. White, of course, will not take the Queen ; 

 probably he would play B to KKt5, to which Black's reply would 

 be Q to Q4, but his game would bo very difficult. 



(.0 To guard against Kt to K5 (ch) and also with the option of 

 jilayiug 1' to Kt4 and Kt5. 



(•}) A miscalculation ; ho ought to have played Q to KR4. 



(/i) Black thought this move saves his piece. White cannot 

 capture tho Knight on account of B to B4, or if 20. Q takes Q, 

 then Kt takes Q, but as will be seen White wins. 



(i) This is the Haw. By being able to take this Pawn with a 

 check he wins tho Knight. 



