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1050 



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Sylvester, James Joseph. 122. Algebraical re- 

 searches, containing a disquisition on NEWTON'S 

 rule for the discovery of imaginary roots, and an 

 allied rule applicable to a particular class of 

 equations, together with a complete invariantive 

 determination of the character of the roots of the 

 general equation of the fifth degree, &c. Phil. 

 Trans. CLIV., 1864, pp. 579-666; Eoy. Soc. 

 Proc. XIII., 1864, pp. 179-183. 



123. On a special class of questions on the 



theory of Probabilities. Brit. Assoc. Eep. 

 XXXV., 1865 (Sect.), pp. 8-9. 



124. Note sur les conditions necessaires et 



suffisantes pour distinguer le cas quand toutes 

 les racines d'une equation du cinquieme degre 

 sont reelles. Paris, Acad. Sci. Compt. Rend. 

 LX., 1865, pp. 759-761. 



. 125. Theoreme d' arithmetique. Paris, 



Acad. Sci. Compt. Rend. LX., 1865, pp. 1011- 



1012, 1121-1125. 

 126. Sur les limites du nombre des racines 



reelles des equations algebriques. Paris, Acad. 



Sci. Compt. Rend. LX., 1865, pp. 1261-1263. 



127. Theoreme d'Algebre e'lementaire. 



Paris, Acad. Sci. Compt. Rend. LXI., 1865, pp. 

 282-283. 



128. On NEWTON'S rule for the discovery of 



imaginary roots of equations. Roy. Soc. Proc. 

 XIV., 1865, pp. 268-270. 



129. On a theorem concerning discrimi- 

 nants. Roy. Soc. Proc. XIV., 1865, pp. 336- 

 337. 



ISO. On LAMBERT'S theorem for elliptic 



motion. [1865.] Astron. Soc. Month. Not. 

 XXVI., 1866, pp. 27-29. 



131. On an elementary proof and general- 

 ization of SIR ISAAC NEWTON'S hitherto unde- 

 monstrated rule for the discovery of imaginary 

 roots. [1865.] London, Math. Soc. Proc. I., 

 1866, No. 2 -, Les Mondes, XL, 1866, pp. 435- 

 437. 



132. On an addition to POINSOT'S ellip- 

 soidal mode of representing the motion of a rigid 

 body turning freely round a fixed point, whereby 

 the time may be made to register itself mechan- 

 ically. London, Math. Soc. Proc. I., 1866, 

 No. 6. 



. 133. Astronomical prolusions : com- 

 mencing with an instantaneous proof of LAM- 

 BERT'S and EULER'S theorems, and modulating 

 through a construction of the orbit of a heavenly 

 body from two heliocentric distances, the sub- 

 tended chord, and the periodic time, and the 

 focal theory of Cartesian ovals, into a discussion 

 of motion in a circle, and its relation to planetary 

 motion. [1865.J Phil. Mag. XXXI., 1866, pp. 

 52-76. 



Sylvester, James Joseph. 134. On an improved 

 form of statement of the new rule for the sepa- 

 ration of the roots of an algebraical equation, 

 with a postcript containing a new theorem. 

 Phil. Mag. XXXL, 1866, pp. 214-218. 



135. Note on the periodical changes of 



orbit, under certain circumstances, of a particle 

 acted on by a central force, and on vectorial co- 

 ordinates, &c., together with a new theory of 

 the analogues to the Cartesian ovals in space, 

 being a sequel to " Astronomical Prolusions." 

 Phil. Mag. XXXL, 1866, pp. 287-300, 380- 

 385. 



136. Note on a "Memoria Technica" for 



DELAMBKE'S, commonly called GAUSS'S theorems. 

 Phil. Mag. XXXIL, 1866, pp. 436-438. 



137. Note on the properties of the test 



operators which occur in the calculus of invari- 

 ants, their derivatives, analogues, and laws of 

 combination ; with an incidental application to 

 the development in a Maclaurinian series of any 

 power of the logarithm of an augmented variable. 

 Phil. Mag. XXXII,, 1866, pp. 461-472. 



138. On the motion of a rigid body acted 



on by no external forces. Phil. Trans. CLVI., 



1866, pp. 757-780; Roy. Soc. Proc. XV.. 1867, 

 pp. 139-144. 



139. On the multiplication of partial 



differential operators. Phil. Mag. XXXIIL, 



1867, pp. 48-55. 



14O. Thoughts on inverse orthogonal ma- 

 trices, simultaneous sign-successions, and tessel- 

 lated pavements in two or more colours, with 

 applications to NEWTON'S rule, ornamental tile- 

 work, and the theory of numbers. Phil. Mag. 

 XXXIV., 1867, pp. 461-475. 



141. On the successive involutes to a 



circle. Brit. Assoc. Rep. XXXVIIL, 1868 

 (Sect.), pp. 10-11 ; Phil. Mag. XXXVL, 1868, 

 pp. 295-306, 459-466. 



142. Address to the Mathematical and 



Physical Section of the BRITISH ASSOCIATION. 

 Brit. Assoc. Rep. XXXIX., J 869 (Sect.), pp. 

 1-8. 



143. Outline trace of the theory of redu- 

 cible cyclodes, i.e., a particular family of succes- 

 sive involutes to a circle whose determination 

 depends on the solution of an algebraico-Dio- 

 phantine equation, and of the number and classi- 

 fication of the forms of such family for any given 

 order of succession. London, Math. Soc. Proc. 

 II., 1869, pp. 137-160, 



144. The story of an equation in differences 



of the second order. Phil. Mag. XXXVII., 

 1869, pp. 225-227. 



145. Note on a new continued fraction ap- 

 plicable to the quadrature of the circle. Phil. 

 Mag. XXXVII., 1869, pp. 373-375. 



