716 REPORT—1896. 
And (c), (C), (C’) into (2), (D), (D’) respectively by writing 2-1 for x. In this 
way the accuracy of the results has been sufficiently confirmed, 
4. Connexion of Quadratic Forms. 
By Lieut.-Colonel ALLAN Cunnincua, R.F., Fellow of King’s Coll. Lond. 
Two quadratic partitions of the same integer (N) are said to be conformal, 
when derivable from one another by mere multiplication by a unit factor, e.g., 
mr? + nv? =1; when not so interchangeable they are said to be 2on-conformal. 
Let N be an integer expressed in two non-conformal quadratic forms. 
N = 60? + mw* = 627 + ny*; (6, m, n integers ; m # n). 
Tien N20, Ary — (yr) 
mw — ny” 
It is shown that a third non-conformal partition may be hence directly com- 
puted by the known processes of conformal multiplication and conformal division 
combined, when 6 is of suitable form, &c. 
i, O= 41; ii. 00,2? +mw,? = +1= 62,7 + ny,?; 
iii, + 00% =mr?—nv*; iv. + 00° =7? —mnv’*. 
Also, in Cases i., ii., iii, any one of the three forms is derivable by the same pro- 
cedure from the other two. 
Ev. N =a? +b? =v? 4+ mw? =2—my’, forms such a Triad that each form is 
directly derivable from the other two as above. 
This is a very useful process for directly effecting a quadratic partition of a 
very large number from two given non-conformal partitions. 
5. On the Plotting out of Great Circle Routes on a Chart. 
By H. M. Taytor, JLA., Fellow of Trinity College, Cambridge. 
It is proposed that on the charts used by ocean-going vessels a series of curves 
should be engraved, each curve representing accurately a great Circle. 
It is shown how such a series of curves may, without the use of mathematical 
calculations, be made use of to plot out on the chart, with much accuracy, the 
Great Circle route between any two points. 
6. On the Stationary Motion of a System of Equal Elastic Spheres in a 
Field of no Forces when their Aggregate Volume is Not Infinitely Small 
compared with the Space in which they Move. By 8. H. Bursury, 
1 FO 
The object of this paper is to prove that the velocities of spheres near to one 
another are correlated. 
1. Consider first the system in which the molecules are material points, between 
which there are no collisions, with their velocities distributed according to Max- 
well’s law. The chance that any molecules shall have component velocities 
Uy Vy + + + Wy is then 
Ae Tew di, G0; - » . AWp, 
and this motion is stationary. 
Let p be the number of molecules per unit volume. 
Let R be a radius at present arbitrary. Definition. Let & ¢ at any point 
P, and at any instant, be the component velocities of the centre of inertia of all 
the molecules which at that instant are contained within a sphere of radius R 
described about P. 
2, If an equal sphere be described about a neighbouring point P’, and P P’ = 3s, 
