208 
and therefore assumes each of these quantities perfect 
differentials, and therefore p a function of p only. 
We will first take fx = 0. 
In any case ^ can evidently be written in terms of the 
normals to the three surfaces (j>, determining the point at 
which (T is the velocity. Thus 
= KiV0i + KaV^2 + KgV03- 
On the previous assumptions Ki, Kg, and Kg must be of a 
definite form, which we proceed to find thus. 
Dt(T = D^Ki. V (/)i + KiD^ V 01 + &c. 
= (d,K. + + K.s|- 
+ &c. + &c. 
( d(T ^ 
DfKi + So-^ J V 01 + &c. + &c. 
= D^Ki. V 01 + AK2. V 02 + D^Kg. V 03 + I V {erf (2) 
whence the required form of the functions Ki, Ka, Kg is given 
D,Ki = ^,D,Ka = ^-^, D,Kg = 
d(j)i 
c?0a 
,(3)* 
^03 
,, dP ^ ^ rr f^P'“ "" 
^ 
where P is a scalar and DjS = 0 2 a scalar function of 
010203 only, and <r takes the form — 
VP + 2l V01 + S 2 V02 + 2gV03 r.(4) 
Proceeding to find Yver, from this we get 
V Vo- = Y( V2l. V01+ V22. V02+ V23-V0g) 
( 
o?2g (iSa'N p p 
- - la + &C. + &C. 
( 5 ) 
^c?0a d(pQ/ 
= Sia + 2a/3 + Sgy say 
where D^2 = 0. 
Whence if the motion is irrotational the S’s can be included 
in the vP. under the above hypotheses. 
Also V V o- may be written H2i.g + K^a.g + H2g.g., and since 
“ &c. denote the edges of the element, the components of 
* Conf. Mr. Hill’s paper on some properties of the equations of hydro- 
dynamics, Q, J. of Maths., Feb,, 1880. 
