PROGRESS OF THE SOLUTION OF THE PROBLEM OF THREE BODIES, 159 
no algebraic integrals, and that it is not possible by any algebraic trans- 
formation which leaves the canonical form of the equations unaltered to 
obtain any further separation of the variables analogous to the elimina- 
tion of the nodes. 
In the second part of the paper (pp. 67-96), the author first, by an 
easy extension of the previous result, shows that no integrals exist which 
involve the time and the variables algebraically, except the known 
integrals, and then finds the integral-equations of the reduced system of 
equations for the problem of three bodies, 7.e. functions of the variables 
whose derivatives with respect to the time vanish when the functions 
themselves vanish ; and shows that the only integral-equation is the one 
whose vanishing expresses the condition that the motion takes place in 
one plane. 
The author then discusses the question, whether any integrals of the 
reduced system exist in the form of integrals of algebraic total differen- 
tials, i.e. the generalised Abelian integrals which have since been studied 
by Picard. This also is shown to be impossible ; and, lastly, this result 
can be extended to the problem of 1 bodies, since, if such an integral 
existed for the problem of n bodies, a corresponding integral for the pro- 
blem of three bodies could be derived by equating all but three of the 
masses to zero. 
A defect in Bruns’s proof (pp. 37 sqq. of Bruns’s paper) was pointed out 
and remedied by Poincaré ! in 1896. 
Gravé? in 1896 showed that the differential equations of the problem 
of three bodies, in the form given by Bertrand, possess no integrals inde- 
pendent of the law of attraction other than those already known ; and 
Painlevé® in 1897-8 extended Bruns’s result, by showing that every 
integral of the problem of n bodies which involves the velocities alge- 
braically (whether the coordinates are involved algebraically or not) is 
an algebraic combination of the known integrals of energy and momentum 
On Solar Radiation.— Report of the Committee, consisting of Dr. G. JOHN- 
STONE STONEY (Chairman), Professor H. McLxop (Secretary), Sir 
G. G. Stokes, Professor A. ScuusTER, Sir H. E. Roscoe, Captain 
W.de W. Abney, Dr. C. CHREE, Professor G. F. FirzGErap, 
Professor H. L. Cattenpar, Mr. G. J. Symons, Mr. W. E. 
Witson, and Professor A. A. RaMBAUT, appointed to consider the 
best Methods of Recording the Direct Intensity of Solar Radiation. 
Tur Balfour Stewart actinometer is now in the hands of Professor 
Callendar, who proposes to employ it in connection with one of his bolo- 
metric methods. 
The Committee therefore asks for reappointment. 
1 «Sur la méthode de Bruns,’ C. &. cxxiii. pp. 1224-8. 
2 ¢ Sur le probléme des trois corps,’ Wowvelles Annales (3) xv. pp. 537-47. 
3 ¢ Sur les intégrales premiéres de la Dynamique et sur le probléme des x corps, 
C. R. oxxiv. pp. 173-6, 1897; ‘ Mémoire sur les intégrales premiéres du probléme 
des n corps,’ Bull. Astr. xv. pp. 81-113, 1898. 
