136 REPORT-—1899, 
numerators may occur, and so the question of convergence is not definitely 
settled. 
The above expansion in powers of « is noticeably similar to that of 
Poincaré. 
SIV. Memoirs of 1868-89 on the Absence of Terms of certain Classes 
from the Infinite Series which Represent the Solution. 
The distinction between the secular and periodic inequalities of the 
elliptic elements of a planet’s orbit has already been explained. Laplace 
in 1773 showed that one of these elements—the mean distance or semi- 
major axis of the orbit—has no secular inequalities at all, when terms of 
higher orders than the first powers of the masses and the squares of the 
eccentricities and inclinations are neglected. Lagrange in 1776 proved 
that this result still holds when all powers of the eccentricities and 
inclinations are taken into account ; and in 1808 Poisson showed that it 
is still true when terms involving squares of the masses are included in 
the calculations. 
In the period under review, Tisserand ! in 1875-6 simplified Poisson’s 
proof by using the transformation of Jacobi and Radau, thus reducing the 
problem of three bodies to a system of the twelfth order, depending on a 
single perturbing function. 
In 1874-5, Mathieu? extended the discussion so as to include terms 
multiplied by the cubes of the masses. He first, by using Jacobi’s sub- 
stitution, replaces the sun and three planets by three fictitious planets 
moving round a fixed centre ; the orbits of these bodies are homothetic 
with the actual orbits, and consequently the study of the variations of the 
mean distances in the fictitious orbits will give the required results. The 
author then, by developing the disturbing function as far as terms of the 
third order in the masses, shows that the reciprocals of the mean distances 
have no secular variations of this order. 
In 1877 Haretu * published an extract from a memoir 4 which appeared 
in 1883. He uses the transformation by which Tisserand had, in 1875, 
simplified Poisson’s work, and discusses a memoir published in 1816, in 
which Poisson believed he had proved the non-existence of secular terms 
in the mean distances, of the third order in the perturbing masses, when 
the variations of the elements of the disturbed planet only were taken 
into account. Haretu shows that Poisson had overlooked a certain class 
of terms, and proves that secular inequalities arise from these terms, 
which are not ultimately cancelled ; and hence that the theorem of the 
invariability of the mean distances holds only to terms of the second order 
in the disturbing masses. Haretu, however, does not give the explicit 
analytical expression of the third order terms in the secular inequalities. 
1 «Mémoire sur un point important de la théorie des perturbations planétaires,’ 
Mémoires de lV’ Académie de Toulouse (7) vii. pp. 374-88 ;y Annales de U'Ecole Norm. 
Sup. (2) vii. pp. 261-74 (merely a reprint); ‘Note sur l’invariabilité des grands 
axes des orbites des planétes,’ C. R. Ixxxii. pp. 442-5. 
2 ‘Mémoire sur les inégalités séculaires des grands axes des orbites des planétes,’ 
C. R., xxix. pp. 1045-9; Crelie, xxx. pp. 97-127. 
8 «Sur Vinvariabilité des grands axes des orbites planétaires,’ C. R. lxxxv. pp. 
504-6. 
4 «Sur Vinvariabilité des grands axes des orbites planétaires,’ Annales de U’ Obs. de 
Paris, Mémoires, xviii. (39 pp.). 
