of Edinburgh , Session 1871-72. 
Gil 
Hence 
1 
K 
_4 
T 
- 1 2 
1 + + -r 
/c A: 
or 
a 
2_^1 
1 + Zs 
Lagrange’s transformation is equivalent to 
2 ^/A; sin. 0 
sin. <p 
1 + k sin. 2 0 ’ 
which, for limits 0 and sin -l i for 0 ) gives 0 and sin —1 -!-for ©; 
k k l 
and we thus have 
sin. -T — 
1 
k. 
dtp 
0 
/ 1 
Jk , 2 ^ 
smrp 
2 k l 
Jk 
sin. 
k 
dO 
0 
Ji ~ sin - 20 - 
whose application to the pendulum problem is obvious. 
5. On the Decomposition of Forces externally applied to an 
Elastic Solid. By W. J. Macquorn Rankine, C.E., 
LL.D., F.R.SS. Bond, and Edin. 
(AbstractJ 
The principles set forth in this paper, though now (with the 
exception of the first theorem) published for the first time, were 
communicated to the French Academy of Sciences fifteen years 
ago, in a memoir entitled u de l'Equilibre interieur d’un Coips 
solide, elastique, et homogene/’ and marked with the motto, 
“ Obvia conspicimus, nubem pellente Matliesi,” the receipt of which 
is acknowledged in the Comptes Rendus of the 6th April 1857 
(vol. xliv. p. 706.) 
The author quotes a theorem discovered by him, and previously 
published in the Philosophical Magazine for December 1855, 
called “ the Principle of Isorrbopic Axes,” viz., cc Every self- 
