218 REPORT—1 862. 
dH dg dH dg 
er dp, dp; 14; 
and guiding himself by a theorem in relation to canonical integrals obtained 
in his memoir of 1855 (see No. 66 of my former Report), he finds by a 
somewhat intricate analysis the expressions of the eight new variables 
Py Por Py» Py Us Ie a> We The results ultimately obtained are of a very 
remarkable and interesting form, viz. H=funct. (p,, P,, Ps, Pys UG» Yo» Ya» Ya) 18 
equal to the value it would have for motion in a plane, plus a term admitting 
of a simple geometrical interpretation, and he thus arrives at the following 
theorem as a résumé of the whole memoir, viz., 
‘In order to integrate in the general case the problem of three bodies, it 
is sufficient to solve the case of motion in a plane, and then to take account of 
a disturbing function equal to the product of a constant depending on the 
areas by the sum of the moments of inertia of the bodies round a certain axis, 
divided by the square of the triangle formed by the three bodies.” 
123. It may be remarked that the only given integral of the system of 
eight equations is the integral of Vis Viva, H=const., and that using this 
equation to eliminate one of the variables, and omitting ‘the equation (=dt), 
we have, as in the solutions of Jacobi and Bertrand, a system of six equations 
between seven variables. As the equations are in the standard dynamical 
form, no investigation is needed of the multiplier M, which is given by 
Jacobi’s general theory, and consequently when any five integrals of the six 
equations are given, the remaining integral can be obtained by a quadrature. 
In the case of three bodies moving in a plane, the solution takes a very 
simple form, which is given in the concluding paragraph of the memoir. 
=0; 
Transformation of Coordinates, Articles Nos. 124 to 141. 
124. It may be convenient to remark at once that two sets of rectangular 
coordinates may be related to each other properly or improperly, viz., the axes 
to which they belong (considered as drawn from the origin in the positive 
directions) may be either capable or else incapable of being brought into 
coincidence. The latter relation, although of equal generality with the former 
one, may for the most part be disregarded ; for by merely reversing the direc- 
tions of the one set of axes, the improper is converted into the proper relation. 
125. In the memoir “ Problema Algebraicum &c.” (1770) Euler proposes to 
himself the question “ Invenire novem numeros ita in quadratum disponendos 
, B, 
D, E, F 
GAEL E 
ut satisfiat duodecem sequentibus conditionibus,” &c., viz., substituting for 
A, B,C, &e. the ordinary letters 
cm a 
Faith note 
fe a; B eS ae: 
the twelve conditions are 
a® +a? +a’?=1, ap+a'p' +a" Bl ’ 
Be +B" +6R=1, By+By'+B"y"=0, 
yore ty =, yaty'al+y'a'=0, 
a +P +7 =1, aa! +B6 +y 
of 
al? +” +y?=1, aia" § pip" +y vay "==(), 
Zp PML, aa” +3" +y"y =0 
