266 Mr. EARNSHAW, ON THE VALUES OF THE SINE AND COSINE 



The first term of this integral has been put equal to zero on the ground that 6 = co would 

 make P = as were this term allowed to remain. (I shall shew presently that it is not allowable 

 to put b = 05). The value of C' is then found by putting 6 = 0, the very supposition which must 

 necessarily render the result erroneous, seeing that dlP - cb'P is then equal to 05 . I infer there- 

 fore that there is no certain ground for writing — for C ; as little indeed as there is for rejecting 



the term Ce"'. In fact, the given function being unchanged when - 6 is written for h, the inte- 

 gral must possess the same property, which gives C = C, and therefore we ought rather to write 



P=C(e'"' + e-'"). 



29. I shall now endeavour to shew that we may not put 6 = eo in the value of P. 



It is easy to shew that 



_ (%axQ,oihcD 06 Sin 6,1'") ir = CO 

 il(aP) - h'iaP) = [-^^^^^ - -^TT^j, = • 



For all Jinite values of 6 the right-hand member of this equation vanishes : but when 6 = co 



the term - Sin (6 . 0) cannot be put equal to zero ; this term corresponds to the limit .r = 0. Also 

 a 



the intermediate steps by which this equation is obtained from P = I — — -^ dx forbid us to 



put 6 = 0. Hence if we put the right-hand member of the equation equal to zero, we are to keep 

 in mind that that step involves a prohibition against putting 6 either equal to zero or oo . Exclusive 

 then of these values of 6, we have 



dKaP) -b\aP)=0; 



and .-. aP = Be"" + B' e-'K 



For the same reason as before, B' ^ B ; and by comparison of this with the value of P (admit- 

 ting that value to be correct for the present), found in the last article we learn that B is inde- 

 pendent both of a and 6, 



... P=-(e"* + e-'"''). 

 a 



How B is to be determined, I know not, seeing that it is not allowable to put 6 = 0, which is 

 the usual plan. 



30. There is great advantage in forming two distinct differential equations for P, as we may 

 learn from one of them something which may assist us in managing the other. In Art. 29, we 

 have seen that, subject to the condition of 6 being finite, we have strictly da(oP) -6^(fflP) = 0; 

 but this condition will not allow us to strike out the right-hand member of the equation in (28). 

 This shews that B and B' in (29) are functions of 6 ; and (29) shews that C, C in (28) are 

 functions of a. 



In strictness then we ought to integrate the equation 



dl(aP) - V{aP) = - ^ Sin (6 M ). 

 6 



C , ai -„6x e-"'' re'"'?>\nib«,)db c°' re-'Sin (605 )rf6 



a a J b a J 



b 

 C being an absolute constant, the value of which I know no means of determining. 



