282 Mr. R. Carmichael on Laplace's Equation, 



is, by an exactly similar process, given by the system 



1 



z + ix+jyz=— 3>(s + 7 1 )+fl 1 s + d 1 



z-ix-jy= —^(« + 7a) + ««« + &. 



<*>"(s + 7i)*"(*+7 2 ) = l 

 where, as before, c„ b lf « 2 , £ 2 are arbitrary constants. 



Thus, by the combination of the system of the previous 

 example with that just stated, we obtain the general representa- 

 tion of the curve of double curvature, whose curvature is constant. 



(III.) To find the integral of the equation of vibratory motion 

 of thin plates, namely, 



. tfF 



Let 



.d .d 



*+•<£+■ 



dt* 



,dx 4 ' dx^.dy 1 dtf 



d*+JTy =J) > 



and the equation becomes 

 d% 

 df- 



+ 6 2 D 4 *=0, 



or 

 Now 



2z=e i v bttf t 0(^) + e -i v bti>* t yfr^y). 



"f dw.e-^-^^^/iT) 

 and putting alternately 



we get for the required solution, 



°°1 dw.e-« li .4>{x + 2iw\/'(i r .bt), y + 2jw\/(-i r .bt)}, 

 s/ir.z= i ■ "" + 



j dw.e-^.^{x-\-2iwj/{— i r .bt) ) y + 2jw>/(—i v .bf)}. 



(IV.) By a similar process the integral of the equation 



d% (d*v d*v d*v d*v d*v d*v \ 



dt* + \dx* "*" dt/* + di* + dy*dz* "* WM " r <fe%*/ 



