OF FLUIDS ON THE MOTION OF PENDULUMS. [101] 



The general equations (8) remain the same as before, but the equations of condition 



become in this case 



v = when t = from x = to at = oo , 



v = V when x = from / = to < = w . 



By Fourier's theorem and another theorem of the same kind, v may be expanded between 

 the limits and oo of a? in the following form : 



2 /»CO /» CO 2 Z* 00 /"CO 



v = - / cos ax cos ax' <b(x', t)dx' da + - / sin aa? sin aa/\J/ (#', J) dat'da. . (184) 



In fact, u could be expanded by means of either of these expressions separately, and of course 

 can be expanded in an infinite number of ways by the sum of the two. If however v had been 

 expanded by means of the first expression alone, its derivatives with respect to at could not 

 have been obtained by differentiating under the integral signs, inasmuch as the derivatives of 

 an odd order do not vanish when at = 0, but would have been given by certain formulae which I 

 have investigated in a former paper.* A similar remark applies to the second expansion, in 

 consequence of the circumstance that v itself and its derivatives of an even order do not vanish 

 with at. But by combining the two expansions we may obtain the derivatives of v, up to any 

 order i that we please to fix on, by merely differentiating under the integral signs. For we 

 may evidently express the finite function v, and that in an infinite number of ways, as the sum 

 of two finite functions (f> (x, t), \J/ (x, t) which like v vanish when x = oo , and which are 

 such that the odd derivatives of the first, and the even derivatives of the second, up to the 

 order i, as well as \|/ (x, t) itself, vanish when x = 0. Substituting now in the second equa- 

 tion (8) the expression for v given by (184), we see that the equation is satisfied provided 



dt M r ' dt * Y 



These equations give 



d) (*?', = x (x) e-»' aH , ^ (x\ t) m o- (*') e ->*' aH , 



where ^, a denote two new arbitrary functions. Substituting in (184), and then passing to the 

 first of the equations of condition, we get 



= x(tf) +"-( a ')> 

 whence <r (at) = — X (•) an< ^ 



«=-/ / cos a(x + x) e'^'^'x (*0 dx'da 

 it J o J o 



(x' + xf 

 VTTfl t J 



The second of the equations of condition requires that 



F- -7=== f «~«7*X(*0 dx m — r f 6 -'' x (2s y/^t) de. 



• On the critical values of the sums of periodic series. Camb. Phil. Trans. Vol. VIII. p. 533. 



