and on Fourier's Theorem. ^Sl 



and cos CO, by showing that the a priori considerations upon 

 which that method rests sometimes conduct us to erroneous 

 results, I shall now proceed to the examination of the specific 

 proofs which have been brought forward to establish the con- 

 trary doctrine. 



In 1781, as we are informed by M. Lacroix, {Traite du 

 Calc. Diff. et du Calc. Integ. tom. iii. 1206), Euler presented 

 to the Academy of St. Petersburg the following formulae : 



r -av v-\^ (;?-l)... 3.2.1 , 1 



/ e ^yyP ^dycosry=i— ~ cospd 



r-,vv-i, ' ( P-I)... 3.2 .1 . \^=o)h^^'^ 

 J e iVyP ^ dysmry—— ~ sin pQ 



where q =ycos 5, r =ysin Q. The same formulae were sub- 

 sequently obtained by M. Poisson by a different method. 

 From these M. Lacroix proceeds (Art. 1210) to deduce the 

 values of the definite integrals 



jdxcosrXi Idxs'mrWf I ~ "^ ), 



the first of which he proves to be = 0, and the second = — . 



r 



Whether these last results are M. Lacroix's own, or are to 

 be attributed to either of the eminent persons above men- 

 tioned, I am unable to learn ; it is my present object to prove 

 that they are erroneous. 



Integrating by parts, we have 



r^-^yvPdu^ '~''y^ p^'^'y""' p.{p-\)s-^yyp-^ 



J y y k k^ F 



- &c. ... - 



p.(jD-l) ...3.2.1s-*y 



a formula which is always true when p is either a positive in- 

 teger or zero, and ^ is not = 0. 



\{ k — q -^ r V — 1 =y (cos 9 + ^ — I sin 5), we have 



fy 



} yP z-iy (cos rj/ — V — 1 sin ry) 

 = —e~9y{cosry— \/ — Isxnry) x 



yp ^ py^~^ 



I /(cos fl + ^ - 1 sin 9) p (cos 9+^-1 sin ^f 



_j(^-l)£2i__ + &e. 

 /^(cos9+ '/-lsind)3 



p. (;?— 1)...3.2. 1 



^fP-^^ (cos9+ V^sin5)^+i* 



