Deep Water due to Motion of Submerged Bodies. 6 1 

 obtained by means of the principle of stationary phase, are 



.costf (54) 



Zzfah^O^'^IZ e 



a 2 a 



(t-rY 



2*7T 



V ft? 



(55) 



The value or t applicable in these expressions is that for 

 which the two values t x and t 2 coincide, namely 



T=7j-(3«— »«)) 



and these expressions then reduce to 



v3 - %A*+K) 1 



3Ma 3 -^ z + h) [l^YQ y 1 



. • • (56> 



&(X,y,,) = - s ,^« -X*- sln {(o2) ! X J 



. . . (57) 

 the motion being now referred to an origin in the free sur- 

 face vertically above the instantaneous position of the centre 

 of the moving sphere. The approximate solution we have 

 obtained for the fluid motion due to a submerged sphere 

 moving with velocity v is accordingly 



a 3 vX a 3 vX 



- 2\x'+f+( Z -hy\i - 2ix*+ } f+{s+hy\t + * 2 ' (58) 



3 7 



where <£ 2 and f 2 are given by equations (52) to (57) above. 



In a later paper we hope to carry the investigation further 

 and to calculate the wave-making resistance of the sphere. 



