On some Identities derived from Elliptic-Function Formula. 539 



oxalic acid : a white heavy powder^ consisting of the oxalates of 

 the earths of the cerium and yttrium group, remained ; this 

 was treated with sulphuric acid, and the yttrium group separated 

 from the cerium in the usual way. Traces only of the cerium 

 group were obtained. The greater mass, consisting of the yttrium 

 group, after conversion into chlorides, showed the absorption - 

 bands of erbium very plainly, and also most distinctly the 

 yttrium-lines in the spark-spectrum. The earths of the cerium 

 group, also converted into chlorides, showed the didymium ab- 

 sorption-bands distinctly, but no recognizable traces of cerium 

 or lanthanum in the spark-spectrum; a tolerably distinct yttrium 

 spectrum showed the deficiency of the separation by the usual 

 method with potassium sulphate. 



LXII. On some Identities derived from Elliptic-Function Formulae, 

 By J. W. L. Glaisher, M.A., F.R.S* 



I. ~ " do not think any one has specially noticed a curious series 

 -L of identities involving the expression 



arc tan q — arc tan q 3 + arc tan q b — arc tan q~ -f &c, 



which follow very simply from some of Jacobi's formula? in the 

 Fundament a Nova. 



Thus take the equation 



1 — a* 1— a 4 l — n 6 \—n 8 



1 q 1 q L q.i q . .. = 1 3 + 6 10 15 



1— q . 1— a?. 1 — q b , 1 — q 7 . . . 22*22 > 



and write therein qi for q ; also let a i = arc tan q } a 3 = arc tan q 6 > 



&c. ; then 



l + g 2 .l- g 4 .l-fg 6 ... 



\/(l + 5 2 ) .\/(l + q 6 ). . .(cos u Y — i sin «j) (cos « 3 + zsina 3 ) .. . 



= I-q 6 -q l0 + q 28 + q 36 -&c. + i{q--q 3 -q™ + qZ i +q* 5 -&c.), 



and 



cos { arc tan q — arc tan q 3 + arc tan q b — &c. } 



l-q 6 -q }0 +q * s + q 36 -&c. 

 = ^/(l + 5 2 ).v / (H-5 6 j...l-r/.l- 5 8 ..7 



sin {arc tan q — arc tan # 3 + arc tan q b — &c.} 



_ q-q s -q Vo -\- q 21 + q Ab - &c. 



" V#+"f)'. V(l + q 6 )...l-q 4 . 1 — g 8 7T7' 



where the exponents in the numerators are the triangular num- 

 * Communicated by the Author. 



