222 REPORT—1862. 
moir “ Nonnulla theoremata generalia &c.” (1775): the geometrical investi- 
gations are given more completely and in greater detail in Lexell’s memoir. 
The result is contained in the following system of formule for the transfor- 
mation of coordinates, viz., if a, 3, y are the inclinations of the resultant 
axis to the original set, and if ¢ is the rotation about the resultant axis, or 
say the resultant rotation, then we have 
x Y | Zz 
ae ot SeeeT Wee 
cos’«+sin?acosp cosacos3(1—cos¢)-+-cosysing cosacosy(1—cos¢) —cosBsing 
cosycosa(1 —cos¢)-+-cosBsing|cosycos8(1 —cos¢) —cosesing \cos?y +sin?ycosp 
I 
Euler attempts, but not very successfully, to apply the formule to the 
dynamical problem of the rotation of a solid body: he does not introduce 
them into the differential equations, but only into the integral ones, and his 
results are complicated and inelegant. The further simplification effected by 
Rodrigues was in fact required. 
137. Jacobi’s paper, “ Euleri formule &c.” (1827), merely cites the last- 
mentioned result. 
138. I find it stated in Lacroix’s ‘ Differential Calculus,’ t. i. p. 533, that 
the following system for the transformation of coordinates was obtained by 
Monge (no reference is given in Lacroix), viz., the system being as above, 
%, ’ p ? ¥ ’ 
Fs B', Mh? 
a, B", y's 
and the quantities «, f’, y’ being arbitrary, then putting 
l4+a+/'+ y'=M, 
1+a—p'—y"=N, 
so that 
M+N+P+Q=4, 
we have ; 
23 =VNP+ ¥ MQ, 2y' =VPQ4+ VMN, 2a" = VYQN+ V MP, 
2g’ VNP_V MQ, 26’=VPQ— VMN, 2y =VON— VME. 
These are formule very closely connected with those of Rodrigues. 
139. The theory was perfected by Rodrigues in the valuable memoir “ Des 
lois géométriques &c.” (1840). Using for greater convenience X, p, v in the 
place of his 3m, 4n, 4p, he in effect writes 
tan 3 cosa=), 
tan 3¢ cos B=p, 
tan 36 cos y=, 
and this being so, the coefficients of transformation are 
14+N—W—r, 2(Au+r) » 2r\.—p) : 
2(urA—v) > 1—’+pe—r*, 2A(uv+r) - 
207A +p) >» 2ru—A » 1—-Nv—p’ +’, 
x 
y \cosBcosa( 1 —cos¢) —cosysing|cos?8+sin*Bcosp cosBcosy(1—cos¢) +cosesing 
Zz 
> 
