204 ^ THIRD REPORT — 1833. 



pressing the various relations between the cosines and sines of 

 multiple arcs and the powers of simple arcs, whether ascending 

 or descending, have been given by Lagrange * and other writers 

 as general, which are either degenerate forms of the coi'rect 

 and more comprehensive eqviations, or altogether erroneous. 

 Poisson had pointed out some of the inconsistencies to which 

 some of these imperfect equations lead, and had slightly hinted 

 at their cause and their explanation ; and the discussion of such 

 cases became soon afterwards a favourite subject of speculation 

 with many writers in the Mathematical Journals of France f and 

 Germany % ; but the complete theory and correction of these 

 expressions was first given by M. Poinsot in an admirable me- 

 moir which was read to the Academy of Sciences of Paris in 

 1823, and published in 1825. They form a most remarkable 

 example of expressions extremely simple and elementary in 

 their nature, which have escaped from the review and analysis 

 of the greatest of modern analysts, in forms which were not 

 merely imperfect, but in some cases absolutely erroneous. 



The difficulties which have presented themselves in the 

 theory of the logarithms of negative numbers, as compared with 

 those of the same numbers with a positive sign, have had a 

 very similar origin. If we consider the signs of quantities as 

 yflc/or,y of their arithmetical values, and if we trace them through- 

 out the whole course of the changes which they undergo, we 

 shall find many examples of results which are identical when 

 considered in their final equivalent forms, but which are not 

 in every respect identical when considered with respect to their 

 derivation: thus (+ aY is identical with {— a)^, when consi- 

 dered in their common result + a^, but not when considered 

 with respect to their derivation. Let us now consider their se- 

 veral logarithms, the common arithmetical value of the logarithm 

 of a being denoted by p : 



Iog(+a)2 = log(l)2a2 = 4r7r \/'~i +2p (1.) 



log (- af = log (- ly^ a^- = (2r + 1)2 nf \/'^l + 2p (2.) 



log a^ = log 1 . o- = 2 /• Tf V' ^ + 2 p (3.) 



It thus appears that the values of log ( + «)^ and log ( — a)- are 

 included amongst those of log «^, but not conversely; and also 

 that the values of log (+ «)^ and log(— a)'^, the arithmetical 

 value being excepted, are not included in each other. 



* Correajjottdence sur I'Ecole Fulyirvlin'iqHo, torn. ii. p. 212. 



■\ In Gergoniie's-yJ«««/pi' des Matlu'inutiqncs, torn. xiv. xv. xvi. xvii. 



I In Creile's Journal fur die rcine uiid aiigewandie Muthemafik. Berlin. 



