'228 Cambridge Philosophical Society. 



December 9, 1844. — On the Values of the Sine and Cosine of an 

 Infinite Angle. By the Rev. S. Earnshaw, of St. John's College. 



It has been usual with mathematicians to write zero as an equiva- 

 lent for both sin jc and cos x when x becomes infinite. The object 

 of this paper is to examine into the propriety of this usage. The 

 inquiry derives importance from its bearing on the general correct- 

 ness of Fourier's theorem for the transformation of functions, and 

 from its affecting the truth of many remarkable results in definite in- 

 tegrals. Certain principles also whicli have been assumed and acted 

 on by Poisson, Fourier, Cauchy and others, in treating of jjeriodic 

 infinite series, are examined, and sliown to be untenable : for exam- 

 ple, it is shown, that as 1 —a; approaches zero, 1 —x -{ x'^ — x^ + . . . 

 ad inf. does not approach 1 — 1 + 1 — 1-f-... ad inf. as its limit ; that 



this last series has not a unique value, and that its value is not — , 



as has generally been argued. It is also remarked that every series 



of the form a^x" -\-a^^xl +...« x^ -{■ ... is discontinuous in those 



terms which are at an infinite distance from the first, unless the co- 

 eflScients tend to zero as n and v tend to oo . The truth of this de- 

 pends on a circumstance which does not seem to have been remarked 

 before, viz. that however small \—x may be, a value of v can always 



be found so large that {\—x)v may be finite, and therefore x", which 

 is equal to (1 — 1 — x)*, is not equal to 1 in the limiL but, to 

 glim, of {\-x)\ ; ^"^, ''!' 



It is lastly proved that sin oo and cos oo are not equivalent to 

 zero, whether we regard them as the results of integration between 

 limits, or as the limiting cases of more general forms. 



February 10, 1845. — On the Connexion between the Sciences of 

 Mechanics and Geometry. By the Rev. H. Goodwin, of Caius Col- 

 lege. 



This paper contains an attempt to determine the ground of the 

 truth of the elementary propositions of mechanics. I'he remarkable 

 analogy between mechanics and geometry suggests the thought, that 

 perhaps there may be something more than analogy, that in fact the 

 basis of the two may be the same. 1'he author endeavours to show 

 that this is really the case ; the ground of the reasoning is, that force 

 is a physical expression of the two ideas of magnitude and direction, 

 of which a straight line is the geometrical expression, and therefore 

 that propositions which are true for one event are true for the other. 

 Hence it is argued, that inasmuch as the giving two sides of a tri- 

 angle gives the third, so that the third may be considered as the re- 

 sultant of the two already given, so if the two sides represent ybrces, 

 the third will still represent the resultant of the two forces already 

 given. 



Reasoning of this kind does not, of course, admit of a very de- 

 monstrative c\\&ra.ci(iv primd facie ; it is the author's design rather to 

 point out a path to the truth, than to assert that he has cleared away 

 every difficulty. 



