Royal Astronomical Society. 145 



into the arrangement ^ ^ 



[In applying the theorem thus analytically formulized^ it is of 

 course to be understood that^ under the sign ^, permutations 

 within the separate parts of a given arrangement, 



^Ji i\R d-^ih ^P+^ ^p+^ ' ' • ^p+''' ^i ^2 • • • ^py ■^^ 



are- irikdmissible, the total number of terms so included being, 

 restricted to n.{n-l) . . . {n-p + l) -. 

 1.2 ... p J 

 I have therefore shown that all the terms arising from the 

 expansion of the products included under the sign of summation, 

 for which the disjunctive identity (^j ^2 • • • 0» = '^i '*/^2- • • '^p 

 does not exist, enter into the final sum in pairs, equal in quan- 

 tity and differing in sign, which consequently mutually destroy, 

 and that the terms for which the said identity does exist together 

 make up the sum ^ ^ ^^^,^, .bobubo . ohnoq-srioa siii 



r«, «2 • • • «n "1 ^ r«l «2 • • • ««1 . ^')«t 



which proves, upon first principles drawn direct from that notion 

 of polar dichotomy of permutation systems which rests at the 

 bottom of the whole theory of the subject, the fundamental, 

 and, as I believe, perfectly new theorem, which it is the object 

 of this communication to establish. 



The theorem may be extended so as to become a theorem for 

 the expansion of the product of any number of determinants, 

 and adapted so as to take in that far more general class of func- 

 tions known to Mr. Cayley and myself under the new name of 

 commutants, of which determinants present only a particulaTj, 

 and that the most limited instance. to tiifli iisrvr 



26 Lincoln's-Inn-Fields, ^'''>'Ry'>*||p 



•' July 22, 1851. ^.^vf 



)n^ 



XXV. Proceedings of Learned Societies. 



ROYAL ASTRONOMICAL SOCIETY. 



i; oiat 'iMiff^ 



April 11,#^N the Measurements of Azimuths on a Spheroid. By 

 1851. vJ Lieut. A. R. Clarke, R.E. 



The author commences his paper with the following words : — 



"It is generally assumed in geodetical calculations, that the sum 



of the reciprocal azimuths of two stations on a spheroid is the same 



as if the stations were on a sphere and had the same latitudes and 



difference of longitude. This is based on Dalby's geometrical prgof, 



