116 G. H. KNIBBS. 
14. Flow in a tube whose right section is an ellipse varying m — 
excentricity and area.—We now consider the case where the — 
sections of a tube are ellipses or very approximately so, the ellipses — 
being however dissimilar. The dimensions of the semiaxes for 
any two sections do not, unless we know how the surface is 
generated, determine the form or area of those intermediate; q 
consequently in regard to these some assumption must be made, 
We may suppose that the terminal ellipses are similarly situated 4 
throughout,! and that straight lines joining the extremities of 
their axes lie wholly on the surface, every section notwithstanding — 
being sensibly an ellipse. Hence we may put : 
By =Butkx and C, =Cm+t+ kx : 
k and &’ being found similarly to (/)§ 12 substituting respectively 
B and C for # in that equation. Then from (g) in the preceding 
article we have 4 
Patt?) In da opi ia q 
= (Jo (Bmt+kx) (Cm +k x)? So (Bm+hx)*(C+ka)) 
The integration of this expression by the usual method of decom : 
position” leads to unsuitable formule. It will be necessary there — 
fore to adopt some value as a mean radius and express the result 4 
in a series of corrective terms, whose magnitude depends o2 a 
variation from the form of a right circular cylinder. Let ] 
Rn=} (Bi+C, + Bo+C2) 
2 pT (B, ay: C,)+(B, ae) C.) 
ee 4 Bm C.=€ 
ed i vs Me 
IR. and c i 
The integration then leads to the following expression 
8ngL 
Pou 21m (1447? eo, 44094 Bho ki =. (26 | 
The f, 6, and ¢ terms are obviously small. If their sum 
denoted by 1+€ then instead of writing this quantity 7 “ 
peewee 
1 If they be not similarly situated the formule are less rigorous: 
correction for a slow rotation of the axes is probably quite insensible- 
2 The integration by the usual procedure for rational fractions | 
formule which are indeterminate in the limiting case, #.¢. when th? 
ellipses become circles. 
