374 Mr. J. J. Sylvester on a Generalization of 



Along with the model of a young cilium presented to the 

 Prague Museum, I left the following remark, translated into 

 German by my honoured friend Prof. Purkinje : — " From ana- 

 logy it appears extremely probable that the heart arises in like 

 manner out of the nucleus of a cell, being originally such a 

 double spiral [as that in fig. 9] . If so, the spiral form of the 

 heart may be explained by the continued division of what was 

 originally a double spiral fibre." 



To the subject of self-division, as part of the process of repro- 

 duction, more importance will by and by be attached than here- 

 tofore. For as the properties of the simplest form of separate 

 independent locomotive Infusoria descend to it from progenitors 

 by fission, — by the same fissiparous mode it appears to me do 

 properties descend from cell to cell, or rather from nucleolus to 

 nucleolus, though these are not separate but combined, and 

 merely parts of a more complicated organism. Having made 

 known my observations on this subject in Mullens Archiv and 

 in former numbers of this Journal*, I have here merely to re- 

 peat, for the purpose of applying it to the suggestion referred to 

 concerning the heart, that the filaments of all organic fibre are 

 made up of particles (nucleoli), and that these particles, and 

 therefore the filaments of fibre, are reproduced in no other way 

 than by self-division f. 



LVII. On a Generalization of the Lagrangian Theorem of Inter- 

 potation. By J. J. Sylvester, F.R.S.% 



THERE is a well-known theorem of Lagrange for determi- 

 ning the form of a rational integral function of one vari- 

 able and of the degree m, when its values corresponding to m 4- 1 

 values of the variable are assigned. M. Cauchy, in his Cours 

 d 3 Analyse de VEcole Poly technique, has extended this theorem 

 to the case of a rational fraction, of which values corresponding 

 to a sufficient number of values of the variable are given ; but 

 the solution of the question there given, although of course 

 correct, is unsatisfactory, as it presents the numerator and de- 

 nominator under forms not strictly analogous. 



The theorem of Lagrange, in respect of its subject matter, may 

 be best generalized as follows. 



Suppose any number of functions of x of the several degrees 

 m x — 1, w 2 — 1, . . . Wf— 1, say U„ V& . . . U„ and that the equation 

 / 1 .U, + / 2 .U 9 + ...+/<. U<=0 



* Mailer's Archiv, Heft vi. 1850. Phil. Mag. August and September 

 1852. 



t See a paper of mine in the Edinburgh New Philosophical Journal for 

 October 1853, " On Animal and Vegetable Fibre." 



X Communicated by the Author. 



