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through zero, hut only by making in imaginary and altering its am- 

 plitude hy it. The author succeeded in determining directly the 

 arbitrary constants for m positive, but not for m negative. It was 

 found that if, in the analytical expression applicable in the case of m 

 positive, — m were written for m, the result would become correct 

 on throwing away the part involving an exponential with a positive 

 index. There was nothing however to show d, priori that this pro- 

 cess was legitimate, nor, if it were, at what value of the amplitude of 

 m a change in the analytical expression ought to be made, although 

 the occurrence of radicals in the descending and ultimately divergent 

 series, which did not occur in ascending convergent series by which 

 the function might always be expressed, showed that some change 

 analogous to the change of sign of a radical ought to be made in pass- 

 ing through some values of the amplitude of the variable m. The 

 method which the author applied to this function is of very general 

 application, but is subject throughout to the same difficulty. 



In the present paper the author has resumed the subject, and has 

 pointed out the character by which the liability to discontinuity in 

 the arbitrary constants may be ascertained, which consists in this, 

 that the terms of an associated divergent series come to be regularly 

 positive. It is thus found that, notwithstanding the discontinuity, 

 the complete integrals, by means of divergent series, of the differen- 

 tial equations which the functions treated of satisfy, are expressed 

 in such a manner as to involve only as many unknown constants as 

 correspond to the degree of the equation. 



Divergent series are usually divided into two classes, according as 

 the terms are regularly positive, or alternately positive and negative. 

 But according to the view here taken, series of the former kind ap- 

 pear as singularities of the general case of divergent series proceeding 

 according to powers of an imaginary variable, as indeterminate forms 

 in passing through which a discontinuity of analytical expression 

 takes place, analogous to a change of sign of a radical. 



A communication was likewise made by the Rev. W. T. Kingsley, 

 " On the application of Photography to Wood Engraving." 



May 25, 1857. 



Mr. Bashforth made a communication " On some Calculations and 

 Experiments undertaken for the purpose of testing the Theories of 

 Capillary Action." 



Also Mr. Candy exhibited a Physiological Alphabet. 



The principle of this alphabet is to make the form of the letter 

 indicate the manner in which the sound is produced, by showing the 

 position of the organs of speech concerned. 



