283] DEVELOPMENTS OF THE DYNAMICAL THEORY. 301 



Then we have ( 279), 



, cos (aco) 



Ct = CO - 



and also ( 278), 



Pi&amp;lt;*i f x p 2 a-2 x p (i a 6 f , 



&&amp;gt;! = cos (&)), eo 2 = ~ cos (CLCO), . . . o&amp;gt; 6 = cos (aw). 



Pa Pa Pa 



But from the fact that to &quot; is the resultant of ?/&quot; and p&quot; we must have by 

 resolving along the screws of reference 



/ / / / /// /// /// /// / / /// // /.\ 



CO C0 1 = T) 77J + P PI, CO C0 2 =T) 1J 2 + p p 2 , . . . CO CO = r) T] ti + p p 6 ...... (1), 



whence we obtain by substitution, 



&amp;gt;!! = W + p &quot;pi, dp. 2 CL 2 = if&quot; I* + p&quot;p2, djOflOe = rj &quot;r} 6 + p &quot;p K . . .(ii). 



If we multiply the first of these equations by^/Su the second by^ 2 /3 2 , &c., 

 and then add, we obtain 



as however p is on the reciprocal system we must have, except when p&quot; = &amp;lt;x&amp;gt; 

 to be subsequently considered, 



d^p^a^^r) &quot;^^. 

 In like manner, 



d2p 1 2 a 1 7i = &amp;gt;/&quot; C3 Vr 

 We shall similarly find 



ivi = r&quot;^&amp;gt; faptfi* = ?&quot;* } 



} ............... (Hi), 



l 1 = r^&amp;gt; 7^ 2 7i#i = ?&quot;&quot;K I 



whence by multiplication 



But we have chosen the intensities rj &quot;, % &quot;, % &quot; so that no one of them is 

 either zero or infinity, whence 



T\P Sy ^ = OT iy OT far^ ........................... (iv). 



It remains to see whether this formula will continue to be satisfied in the 

 cases excepted from this demonstration. 



Let us take the case in which p^ is infinite, which makes ^ ... infinite. 

 We have in the case of p^ very large, 



ffr, cos (771) 

 * = ~2p^&amp;gt; 

 the equations (ii) become 



TJ&quot;p^ cos (771 ) . T; &quot;^,, cos (776) 



= Z = 



