in Laplace's Theory of the Tides. 241 



noticed (§§ 15-17). Remark now that as e (which is essentially 

 positive in the actual problem) is increased from to + go , each 

 of the ratios R,, R 2 , . . . R^ R, +I , . . . increases from zero, each 

 one more rapidly than the next in ascending order, until R t 

 becomes -f- go , and suddenly changes to — oo , and again goes on 

 increasing till it again reaches -f co and suddenly — oo , and so 

 on. But before R t becomes go the second time, R 2 becomes + oo , 

 — go , and again increases towards + go . The same holds for 

 each of the other ratios ; that is to say, as e increases continu- 

 ously, each one of the ratios is always increasing except when its 

 value reaches +co and passes suddenly to — go . The order in 

 which the values of the different ratios pass through oo is a sub- 

 ject of great interest and importance which requires careful ex- 

 amination. I hope to return to it, and meantime only remark 

 that the formula (8) for calculating R$ from R i+1 shows: — 



(1.) That no two consecutive ratios can be simultaneously ne- 

 gative. 



(2.) While R J+1 increases from -co to 0, R* increases from 



e 

 to a value somewhat (but very slightly) greater than —. — ^r, and 



goes on increasing till it reaches co , when R t - +1 = « 



2i 4- 1 

 (3.) WhenR i+1 is >jrj-. — s\, and therefore when Rj +1 >l, 



R t is negative. 



From (1.) it follows that in the series of coefficients 



K 2 , K 4 , K 6 , ... 



there cannot be two consecutive changes of sign. From (3.) it 

 follows that each coefficient is less in absolute value than its pre- 

 decessor if of the same sign, except when the predecessor is of 

 opposite sign to the coefficient preceding^; and of two coeffi- 

 cients immediately following a change of sign, the second may be 

 less than the first, but if so, only by a very small proportion of 

 the value of either ; but through nearly the whole range of values 

 of e for which there is a change of sign from, say, K* to K t+1 , 

 K,-+2 is >Kj +1 in absolute value. (For illustration of this see 

 Laplace's series above, for his case of e=10, for which he gives 



K 2 =l, K 4 =20-1862, K e = 10-1164, K 8 = -13-1047, 



K 10 =-15-4488, K 12 =-7'4581, K l4 =-2'1975 ... &c.) 



20. Laplace's brilliant invention which forms the subject of 

 this article is capable of great extension, as I hope to show in 

 a future communication. I have not hitherto found any trace 



Phil. May. S. 4. Vol. 50. Is T o. 330. Sept. 1875. R 



