Dec. 29, 1882.] 



• KNOWLEDGE 



501 



hy /'ydx 



have seen, is represented 



(Suppositions the same as before.) Tlien drawing Ad, B L, per- 

 pendicular to y, we might find it conyenient to regard y as the 

 dei)cndent variable, passing from d to L, instead of taking x 

 from a to M. As before, we should get for our element of area 

 P n ; where N n is Ax, a small increment of x. We can, however, as 

 readily use 1 K, or Ay, the corresponding increment of y. For wo 



have Xn = Pk = ?J^ . qk=J^ . Ay. 



q k Ay 



That is rectPn = PN . Pk = y^. Ay ; 



Ay 

 or ultimately when A.r, Ay are made indefinitely small, 

 T> di 



d,j 



and taking y from O d ( = fli supp( 

 area A 



I have 



) to OL( = fc, suppose) 



Sia=/'"'y'll.dy. 

 J «i dy 



In any such case where we have to integrate between x. = a and .t = fc, 

 the expression Indx, where u is some fraction of x, we may regard 



X as a function of some new variable y, if only by changing ij 

 between certain limits a, and bj we get all the values of « which it 

 would receive when we changed x between a and b. We have then 



«<?!■ = »^ . dy and /' wdx = /"■«!?£ . dy 



and the work of integration is reduced to that of integrating the 



indefinite integral /u~ . dy. 



J dy 

 Suppose, for example, we had to integi-ate 



fz 



x^x 



Let us try the experiment of putting x = ~. 



Then g; = -land/' "^^ _/ ^'^ /--^Vv 



^y y' J xs/x'-d- J j\-a-xi\ r' 



Again, take / 



J X./-2. 



:= — -sil 



denominator may be written X\/3?—{x, 

 putting ^Ili^=z, which will obviously si 



Here noting that the 

 1}-, wo are led to try 

 )lify the radical. 



This gives 1 — - = z, ( 



•/ xJtax-x' J x' 



dz oJ J\-z' 



v/1- 



: sin~'a =- sm 



This method of integration by parts is usually tentative, — several 

 substitutions may be tried before one is hit upon which gives an 

 integrablc expression. 



IXTECRATIOX UY PAKTS. 



Another method (also tentative) is available, when we are 

 endeavouring to integrate an expression. 



Supposing we l>ad to obtain the area A B 51 a. Wo might in 

 some cases find it more convenient to determine instead the area 

 A B L d, which gives us what we are seeking, because 



area A B M a = rect L M — rect d a — area A B L d. 

 If wo regard ON, PN (or P K, PN), the ordinates of P, as 

 functions x and y of some third variable (, then wo havo 



rect K N =a;y = i(, say, 

 and if we increase t by A' so that x becomes On or .r+ Ax, and y 

 becomes 1 or y + A y. Wc havo 



increase of rect K N = rect P n + rect I P (ultimately) ; 

 that is, A(xy) = yAa: + ii'A!/ 



AJ 



aking At indefinitely small 



it dt dt 





whereftrj 



(This really corresponds only to saying that in passing from 

 1^ to B, the increment of rect K N, namely the gnomon L P M, 

 is equal to the sum of the increments of the areas A P N a 

 andAPKd). Thus if an integral can be written in the form 



both functions of t, we may sabstitnto 



/ 



-=^<J', where x and y a 

 for the integral so written 



This is equivalent to finding the area AB L d, instead of the area 



As an example consider the integral 



yiogxd.. 



-do 



This mav be written /—. log xdjc 



J dx 



Hence by the formula just obtained we have 



/log X dx = X log X — /c . -dx 



^vx Cf)C2is Column, 



By Mephisto. 



THE following lively game was played by Mephisto against an 

 amateur : — 



EVANS GAMBIT. 



White. 

 Mephisto. 

 PtoK-1 

 Kt to KB3 

 B toBJ, 

 P to QKt-i 

 PtoB3 

 P toQ-i 

 Castles 

 P takes P 

 KttoB3(r) 

 P toK5 

 B to R3 

 BtkP(ch)(,( 



Blacli 



P to K4 

 Kt to QB3 

 B toB4 

 B takes P 

 B to B4 (a) 

 P takes P 

 P to Q3 (b) 

 B to Kt3 

 Kt to B3 (c!) 

 P takes P 

 Kt to E4 (e) 

 ) K takes B 



White. 

 Mephisto. 



13. KttksP(ch) K to K sq. (y) 

 11. RtoKsq.(;i) B to K3 



15. Pto Q5 Kt takes P (t) 



16. QtoB5(ch)(j)P toKtS 



17. Kt takes P Kt to B3 (i) 



18. RtksB(ch) KtoB2 



19. Kt toK5(ch) (OK takes R 



20. Q to R3 (ch) K takes Kt 



21. RtoKsq.(ch) K to Bt 



22. QtoKt3(ch) K to Bo 



23. K to K5 mate. 



NOTES. 



(«) B to 1) 1- is considered a more reliable defence than B to Rt. 



(/.) liis best. 



((■) Other good eontinnations are either 9. B to Kt2 or 9. P to 

 Q5. The game would proceed as follows :— If 9. B to Kt2, KKt to 

 .K2. 10. Kt to Kt5, P to Q4. 11. P takes P, Kt to R I. 12. P to 

 QC, Kt takes B. 13. P takes Kt, Q to Q-1. 14. Kt to QB3, 

 Kt takes B. 15. Kt takes Q, Kt takes Q. 10. R takes Kt, .tc. 

 If 9. P to Q5, Kt to Rl. 10. B to Kt2, Kt to KB3, kc. 



((!) Black may also reply with 9. Kt to R-t, or 9. B to Kt5 



((•) Now Black lays himself open to a strong attack. Had ho 

 played either 11. B to Kt5 or B takes P, White would have played 

 12. Q to Kt3. 



(/) A sacrifice promising a strong attack. 



(,;) This move practically decided the game. Amateur ought to 

 have played K to Kt sq., which would have given him a safe game. 



(/i) Threatening to win the Queen by discovered check with tho 

 Knight. 



(i) Black might have played 15. P to B3 with tho intention of 

 oxihanging Queens if White took tho Bishop at once, which, how- 

 ever, ho would not do, but play Q to R-t first, to bo followed by 

 QI{ to Q sq. Also 15. P to B4 commended itself, but Whito 

 would likewise play Q to Rl (ch), forcing the Black King to play 



