4 REPORT — 1861. 



PuP^iPs ; and A" the distance of the centre of the circle through the feet otp^, p^fP^ ; 

 then by what precedes, 



2A+S'=a+i3+y; 2a" = A+S'; 



4A" = 8' + «+/3+y; and 



s a b c 



Let A,, Aj, A3, A'„ A'2, A'3, be the distances fi-om the plane of reference of the 

 centres of the cu'cles circumscribing the component triangles of the complete qiuidii- 

 laterals of which 0, O^ O^, O3, are the vertical points ; then 



2Ai=8,+S3, &c., 2a/=8 + Sj, &c. ; 

 .-. Ai + A, + A3 + A/+A,'4-A3' = 6A. 



Equating two values of A', we have 



§1, S, ,83^ -8+8.+8,+a 3 

 Pi Vi Pi 2r 



Let_ ^j=^j, §,=;>„ S3 =;,3, (C) 



and this becomes 



Pi I P2 I P3 Pl+Pl+P3 _^Q-Q 



Again, let 



«!=«, K=h ^=<= (D) 



and the same formula gives 



Pl Pl P3 ^''J »"2 '"3/ ^ 



An Inquiry into ilie Fundamental PrincipJes of Algebra, chiefly with rer/ard 

 to Negative and Imagirutry Quantities. By C. F. Eeman. 



On Befinite Integrals. By Bieeexs de Haau. 



On Geometncal Rests in Space. By Sir W. E. Hajiiltok, M.R.I.A. 



On the Boots of Substitutioiis. By the Eev. T. P. Exrkman, M.A., F.R.S., 

 Honorary Member of the Literary and Philosophical Societies of Manchester 

 and Liverpool. 



The group given at page 6 of the " Transactions of the Sections of the British 

 Association for 1860 " is one of the equivalents of a gi-ouped gi-oup, which is of 

 the class described by M. Camille Jordan in the 4th chapter of his These " Sur le 

 Nombre des Valem-s des Fonctions " (Paris, Mallet-Bachelier, 1860, or Jom-nal de 

 I'Ecole Polytechnique, 1861). The fh'st foiu- substitutions of that group foi-m one 

 of the gi'ouped groups described by Cauchy in his " Meuioire sm- les Arrangements," 

 &c. (Exercices d Analyse et de Physique "Mathematique, tome troisieme). 



M. Joi-dan's gi-oup is formed by writing in the auxiliary group 



1234 

 2143 

 3412 

 4321, 



for 1, 2j^ ; for 2, ^^ ; for 3, g- ; for 4, g-,. 



This gives a grouped group G, from which is obtained the equivalent 



G' =34127856 G 34127850, 

 in page 6 quoted above. 



The two gi'oups of ill'. Caj-ley at page 5 of the same Report, are gi'ouped gi'oups 



