IN THE NEIGHBOURHOOD OF A CAUSTIC. 597 



whence rC„^., =—£ sin 9 + -^" cos 6, 



dS„ . ^ dC„ 

 rS„,, = ~sme-~cose, 



from which it easily appears that for what value soever of n the equations first given are true, they 

 remain true when that value is increased by a unit. And that they are true when n = I h proved 

 by common integration by parts. 



" If instead of w we write w'', k being positive, and then for kn write n, we have 



„ e-"°^^«'*.cos(rsine.w*).w'-'dw = -r,„, . cos— . r"S 



k (j) k 



7ftff) 1 fin 1 



^-rco,9...> g;„ (^ gj^ . jt,*) _ j^-i a,v =. -r „ . sin — . r"*. 



If r = 1, and we call these integrals C„ and S„, let us take 

 cos (sin 6 . w^ - m w) = cos (sin &.»'). 1 1 h . . . 



+ sin (sin 9 . w') . [mw - 



2.3 

 Multiplying by e''^"^"'"^ .dw, and integrating, we have 



^»^-C0s9.«P cos (sine. W'- OTM;)dM,= Ci + ^2.W« - Cj '^*T~Z'^ 



"If we now make 9 = -, and observe that in this case C„ vanishes whenever n is an odd 



2 



multiple of 3, and S„ whenever n is an even multiple, we obtain 



/„ cos (?/)^ - tnw)dw = Ci - >Si C, — ;; + A,, 



'2.3 '2.3 6 '"2.3 9 



' "2.3.4 °2.3 7 2.3 10 



2.3 ... 6 



= - r, cos - . - — r, sin - . - . r» cos - . - 



3-1 Is 2/ 3 -a Vs 2^ 2.3 3 1 Vs 2^ 



1 /2 ttN 1 „ /5 7r\ m" 1 _ . /8 ttN m' 



^ 3 ^1 ^'" (i-i) "^ -^ 3 ri*=°n3- 2) -^TiTj - i ^1^'" li-i) 2T3— 7 - 



1 TT , 1 m=* 4 1 m*^ 7 4. 1 m" 



- r r*»s — ' I — + — . — . — . — . — . 



3 i. ()■' 3' 2.3 3 3 2.3... C 3 3 3 2.3... 9 



1 7r,2OT' 52 m' 852 m"> , 



+ -r..cos-.}m--.^-^ + -.-.^-y— ^--.-.-.g 3_j„+-.5- 



" I may observe that the precautions which I have taken, to shew that the algebraical cases 

 limits of aritlimetical ones, are not absolutely necessary in this instance. For if we resolve 



are 



Vol. VIII. Part V. 4H 



