SECT. II.] PARTICULAR TEMPERATURE-VALUES. 233 



A==-2-versinfo-V 



m \ nj 



T o ^ /-i ^ 

 /*, = 2 versin 1 - , 

 7&amp;gt;i \ n) 



H C\ &quot;* I Ct &quot; \ 



i = 2 versin 2 - , 



Z- 



i 1 v s-\ *v I/ tv&quot;! 



1 1} = - 2 versin J (n - 1) - } . 



771 



Suppose then that we have divided the semi-circumference TT 

 into n equal parts, and that in order to form u, we take i of those 

 parts, i being less than n, we shall satisfy the differential equations 

 by taking a l to be any quantity whatever, and making 



sin u sin Qu - ? versin M . 



= .- = e m , 



sin u 



p Sin 2 It Sin Iw -^versinu 



1 sin u 



sin 3i sin 2u ~ versin 



7 = a, : - e 



sin w 



sin ?m sin (n V}u -^ versin w 



w = a. : ^ J e m 



sin u 



As there are n different arcs which we may take for u, 

 namely, 



A 7T -7T 7T , TN&quot;^ 



0- , 1 - , 2 - , , (n i) - , 



71 71 W X 71 



there are also n systems of particular values for a, fS, 7, &c., 

 and the general values of these variables are the sums of the 

 particular values. 



254 We see first that if the arc u is nothing, the quantities 

 which multiply a, in the values of a, j3, 7, &c., become all equal 



., . sin u sin Oz , .. . . 



to unity, since : takes the value 1 when the arc u 



sin u 



vanishes; and the same is the case with the quantities which are 



