Mr. Moon on the Symbols sin oo mid cos oo , ^c. 483 



made as well by this process as by the comparative pendulums, 

 to determine the laws according to which gravity varies 

 throughout the whole extent of the terrestrial spheroid. 



I have performed alone the numerous numerical calculations 

 which are given in this volume and in the preceding one. 

 Although I have made them with care, and most of them se- 

 veral times, I cannot be sure that some errors of detail have 

 not escaped me. But the attentive reader, who must consider 

 them only as examples, will, as he proceeds, himself find op- 

 portunities of correcting them. I have more reason to fear 

 the errors which may have escaped me in the analytical for- 

 mulae, on account of the general influence which they might 

 have upon the applications of them. But a person well-versed 

 in analysis, and very skilful in discovering the errors of the 

 expressions which it employs, M. Yvon Villar^eau, has had the 

 kindness to revise attentively the formulae which this volume 

 contains, and to point out to me the inaccuracies which had 

 escaped in the printing. I have rectified those which were 

 reparable, and which might most easily have misled. I have 

 indicated in an erratum others which, notwithstanding their 

 obviousness, might embarrass the reader for a moment, re- 

 questing him to take the trouble to correct them before read- 

 ing the work, and especially before making any applications 

 of it. 



LXIX. On the Symbols sin oo and cos go, and on Fourier's 

 Theorem. By R. Moon, M.A., Fellow ofQtieen's College^ 

 Cambridge, and of the Cambridge Philosophical Society'^, 



THE remarkable symbols sin oo and cos oe have for some 

 time attracted the attention of mathematicians, and very 

 opposite opinions have been entertained as to their nature. 

 It appears to have been taken for granted in the first instance, 

 that when considered numerically they denote indefinite mag- 

 nitudes, that indefiniteness being confined between certain 

 limits, namely plus and minus unity; and this is certainly all 

 that can be said of these symbols a priori. But recent writers 

 have supposed that the above conclusion is erroneous, and that 

 even considered numerically these symbols have definite 

 values, or in fact that they denote evanescent quantities. To 

 leave undecided a question of this nature would be a stigma 

 on mathematical science. If we adopt the first opinion, we 

 are met by certain analytical difficulties which have never yet 



* Communicated by the Author. 

 2 K2 



