546 Prof. Challis on the Mathematical Principles 



the direction of oe is aPoc ; that in the direction of y is a> 2 y \ and 

 parallel to the axis of z it is zero. 



Now let X, Y, Z be respectively the sums of the impressed 

 forces parallel to the axes of coordinates. Then, from what has 

 been shown above, 



7-7 + dn - 



whence it follows that Xdx + Ydy + Zdz is an exact differential, 

 which for brevity I shall call (d¥) . 



This being understood, we have, in the next place, by the 

 third general equation of hydrodynamics, 



w=( x -(S) &+ ( Y -©)^ + ( z -©)^ 



in which u, v, w have the significations defined in the first 

 paragraph of this communication, the motions under consider- 

 ation being relative to the earth supposed fixed. From this 

 equation it is readily seen that 



Hence if, as is usually done, the investigation is restricted to 

 the first powers of u, v, and w, it will follow, since the right- 

 hand side of this equality is an exact differential, that 



du 7 dv 7 dw , 



7t dx+ Tt d y + ii dz 



is also an exact differential. To satisfy this condition it is 

 necessary and sufficient to assume that udx + vdy + wdz is an 

 exact differential ; for in that case, supposing (d<f>) to be the 

 differential, 



du__d*cj> dv__d*cj> dw_d 2 cj> 

 dt ~~ dxdt* dt """ dydt' dt ~~ dzdt' 

 and 



du 7 dv . dw 7 / 7 dd>\ 



it dx+ di dy+ ii dz ={ d -dt} 



Similarly, if u', v\ w'he the resolved parts of the tidal motion 

 at the same position due to the sun's attraction, we should have 

 u!dx-\-v'dy + w'dz an exact differential. If, therefore, U, V, W 

 be the resolved velocities due to the combined action of sun and 



