164 



THE ROYAL SOCIETY OF CANADA 



du 



dv 



= 2(Z72F3'+f/3F2'), 



= 2(f/2'F3+ f/a'Fs), 



W=^^.etc....) 



Thus equations (22) become 



(220 





= [Fi+2Ci(^2F3'+t/3F2')]. 



If these values of , — and — ^= be substituted in either of the 



V \ <f) 



relations 



-^ ( VT r) = -2^1 r', -^ ( v^Tf) = - 24 f , 



there results an equation which can be written 



(25) -^/= [(t// + è ^ f/i) t/- C/' t/i+2ci n/IF Î/2] 



+ Fi VF" + 2ci FF3 - 2ci F2 



i.e., in the form 



Hu)+fx{v)-\-p{u)cT{v)=0. 

 Hence 



y{u) -\-p'{u)a{v) =0, 



,x'{v)+p{u)a'{v)=0; 

 and therefore p(î^) or (t{v) must be a constant. 



But (t(v) = F and F' 4= 0; the equation (23) therefore implies that 



C3 f/ - t/i x/TF + 2ci ( f/2 - t/ f/s) + Fi s/'F - 2ci ( F2 - F F3) + C3 F = 0, 



where C3 is a constant. From the second of these relations il follows 

 that 



CzU-Ui JW + 2ci{U2 -UUs) = - Ci, 



^3 F + Fi n/~ - 2ci( F2 - FF3) = + C4, 



where C4 is a further constant. 



