572 Mr. stokes, on THE CRITICAL VALUES OF 



by which f is developed, or, which is the same, suppose /i(j/') or f.,(y') to be zero except for values 

 of the variable infinitely close to a particular value y, and divide both sides of the equation by 



IMy')dy' or !My')dy'. 



We get thus from the first two lines of (90) and the first two of (92), supposing y and y positive, 

 and y' the greater of the two, 



S -z ^ , r COSXV cos AW 



2\/3 + sin2X^ e^'-e-^" ^ 



where the first 2 refers to the positive roots of (85), and the second to positive integral values 

 of n from 1 to 05 . 



Of course, if y become greater than y', y and y' will have to change places in the second side 

 of (93). This is in accordance with the formula (92), since now the second line does not vanish; 

 and it will easily be found that the first and second lines together give the same result as if we had 

 at once made y and y change places. Although y has been supposed positive in (93), it is easily 

 seen that it may be supposed negative, provided it be numerically less than y'. 



The other formula above alluded to is obtained in a manner exactly similar by comparing the 

 last two lines in (92) with the last two in (90). It is 



2 7: : ?, Sin 1x1/ sin M« 



2^/3- sin 2;u/3 t^'-e-"' ^ 



= -^- ' ,, Jb 7 ;^^ Y^ 5.sinw.r (94), 



TT (w + A)e"^ + (w - A)6-^ 



where the first 2 refers to the positive roots of (89), the second to positive integral values of w, 

 and where x is supposed to lie between and tt, y' between and /3, y between and y\ or, it 

 may be, between - y and y. Although ,v has been supposed less than tt, it may be observed 

 that the formulae (gs), (94) hold good so long as x, being positive, is less than Stt. 



■18. Let it be required to determine the permanent state of temperature in a homogeneous 

 rectangular parallelepiped, supposing the surface kept up to a given temperature, which varies 

 from point to point. 



Let the origin be in one corner of the parallelepiped, and let the adjacent edges be taken for 

 the axes of .v, y, x. Let a, b, c be the lengths of the edges; /,(?/, z), Fi(y, x), the given tem- 

 peratures of the faces for which w = and w = a respectively ; /„(;y, x), F.j(z, .r) the same for the 

 faces perpendicular to the axis of y; /3(ir, y), F^{x, y) the same for those perpendicular to the 

 axis of X. Then if we put for shortness v to denote the operation otherwise denoted by 



d- d' (F 

 dx' dy" dx- ' 



as will be done in the rest of this paper, and write only the characteristics of the functions, we 

 shall have, to determine the temperature 71, the general equation y ?< = with the particular 

 conditions 



II = f\, when a; = ; u = F^, when a; = a (9.'5) ; 



u = f:,, when y = ; u = F.^, when y = 6 (S^*) ; 



u = fn, when ?r = ; u = F„ when « = c '(97); 



