606 MR THOMAS STEPHENS DAVIES ON 



NOTE B. 

 ON LEMMAS I, II, III, pp. 578-81. 



I am not aware that the general forms of the co-efficients of the multiple cosines have 

 been given by any author ; and, indeed, since they are not required in any class of inquiries 

 which fall under the general objects of science, it is scarcely likely that they should have been 

 an object of research. Cases, however, having much analogy to them, are sufficiently well 

 known ; and the method by which they are treated, naturally pointed in the direction by 

 which these could be obtained when occasion called for them. 



By taking r^=r in Lemma i., and p = r in ii., and comparing the terms in the two cases 

 clear of the cosines, with the corresponding ones in iii., we obtain two elegant formula, first 

 given by EULER in the Acta Acad. Petropolitance, 1781, viz. : 



2(2n-l)(2n-2) ____ (n + 1) 

 tf + *? + tf + ..-. + f _!* + *.' = - i . 2 . 3 .....' 



11.3 1.3.5 2 M (2tt 



2 2 274 4+ 2.4.6 6+ '" = 1.2.3 



NOTE C. 

 ON PROPS. IX, X, p. 582. 



This theorem is well known, and has been demonstrated in a great variety of ways, as a 

 property of the centrdid ; though, probably, it has not been before considered in the light of 

 a porism, nor, consequently, investigated as such. It was suggested by CARNOT (Geom. de 

 Pos. p. 326), that it would offer some advantage to take this point as the origin of co-ordinates ; 

 and as it will illustrate the manner in which, under particular circumstances, the porismatic 

 proposition may become an ordinary indeterminate, the same property is here investigated, 

 with the centroid taken as origin of polar co-ordinates. 



j Z 2 =aj r 2 2^ rr l cos (6 



cos 



a m .A m Z 2 = a m r*-2a m rr m cos(d-d m ) + a m rJ ! 



Now, since the origin is the centroid, the middle vertical column on the right side is 

 zero ; and hence 



S (a. . A m Z 2 ) = S a m . r* + S (a. O 



NOTE D. 

 ON PROPS. XI, XII, p. 583. 



These may be proved, as, indeed, most of the earlier propositions can be, very neatly, 

 but at greater length, by means of rectangular co-ordinates. 



Let a, /3j, 02 &, . . . . a m /3 m denote the given points, xy the arbitrary point Z, a /3 the 



