228 



Mr Sharpe, Liquid Motion from a Single Source 



expand (say) in a series of sines true from to tt/2, then in 

 another series of sines true from to it and subtract one from the 

 other. We shall thus get zero, and we can take 



$ (a m sin mO) = /3 



22 (- 1)*» S i^ + 2 (- 1)- S - i ^~ 



.(14). 



This will be zero from to 7r/2 excluding 7r/2 itself. Instead 

 of expanding in a series of sines &c. we could of course similarly 

 expand any function of 0. Again, by Fourier's Theorem it can 

 be shewn that between the limits = and = 7r/2 (but excluding 

 the actual limit 7r/2) 



2 °° 

 sin(2p + l)0 Xsin2n6. 



IT J 



sin (2ft -2p — l)~ sin (2ft + 2p + 1) ~- 



2n — 2p — 1 



2n + 2p + 1 



= 



(15), 



where p is an integer and 2 signifies summation with regard to n 

 from 1 to oc . By this means, between the given limits, the sine 

 of any odd multiple of 0, or any finite expression of the form 



A sin + B sin 30 + G sin 50, 



can be expanded in series of sines of even multiples of 0. We 

 thus might use the single series (15) with a particular value of p 

 between = and = a x where a x < tt/2, or if we wish to use 

 expansions that would be true even at the limit = tt/2, we might 

 use expansions of expressions like A sin + B sin 30 + G sin 50 

 with the condition A — B + C = 0. Again, to satisfy equation (5) 

 we might take any linear combination of the expressions (13), (14) 

 and (15) multiplied by constants. (5) can therefore be satisfied in 

 an infinite number of ways, so the problem proposed admits of an 

 infinite number of solutions. 



6. We will now proceed to examine in some detail the sim- 

 plest case of equation (5) viz. 



Suppose 



S (a m sin m0) = bt (- l) w+1 sin (2ft - 1) = (16), 



n having all integer values from 1 to oo, and (16) being true 

 from = to = 7r/2 {it\2 being excluded). Equation (8) for 

 finding a x will then become 



7 fsin 2ai 

 fj,7r = c sin a-L + fia! + b { — - — 



sin 4aj sin 6a! _ „ 

 4 6 



.(17). 



