1888.] in a Inquid when the impulse reduces to a cowple. 281 
ie. to a doubly periodic function of the second kind having one 
simple pole. The integral of this function appears to be unknown. 
We may remark that as the function to be integrated has one 
simple pole and is in all parts of the plane at a finite distance 
similar to a rational fraction, the integrated function will have a 
logarithmic critical point w= 0, and will be a uniform function of 
wu in every region of the plane at a finite distance and not in- 
cluding this point. 
13. To sum up.—The angular velocity and orientation of 
the solid at any time are completely expressed. Hach component 
of the angular velocity is proportional to a doubly periodic 
function of the second kind of an argument u proportional to the 
time, and all the constants are well defined. The direction- 
cosines of three axes fixed in the solid with reference to axes 
fixed in space are also completely expressed, each of them as 
a sum of terms of this form. The distance traversed by the solid 
in a direction parallel to one of the fixed axes of reference, the 
axis of the resultant impulsive couple, is in like manner com- 
pletely expressed as a sum of terms of which three are pro- 
portional to u, three are proportional to fu, and three are doubly 
periodic functions of the second kind. The distances traversed 
by the solid in two directions at right angles to this are reduced 
to quadratures. The velocity of the solid in any direction at 
any time is completely expressed; in the case of a fixed direction 
it consists of a sum of terms each of them a doubly periodic 
function of the second kind having one double pole. 
(2) On Prof. Miller’s Observations of Supernumerary hain- 
bows. By J. Larmor, M.A., St John’s College. 
The theory of the supernumerary bows which accompany the 
primary and secondary rainbows has, it is well known, been placed 
on an exact mathematical basis by Airy *. 
A series of observations on narrow jets of water were made 
soon after by the late Prof. Miller+ with the object of comparing 
the magnitudes involved with their theoretical values. But so far 
as appears the comparison was never completed, although the most 
difficult calculation connected with it was supplied by Prof. Stokes. 
Prof. Miller contented himself with giving tables of his observations 
and pointing out that the relative positions of the first few diffrac- 
tion fringes agreed fairly with the indications of theory. 
The rule given by Airy to determine the absolute magnitudes 
of the bands which accompany the principal bow for homogeneous 
* Camb. Phil. Trans. Vol. v1. p, 79. + Camb. Phil. Trans. Vol. vit. p. 277. 
