TRANSACTIONS OP THE SECTIONS. 3 



and we have therefore 



73 174 74 rfi 



i^e— ,7 • „4 '74 74 '-'t-lV'eJ' 



and consequently 



Q.=4^- i^ fK- s{(D,_,(^,)>.(.-;i,)}, 



^,- ^i_4 ^W + 1 



which is the general equation connecting the form of each quotient with that of the 

 denominator to the immediately preceding unreduced convergent in the expansion of 



"— under the form of an improper continued fraction. 

 /« . . 



If instead of the denominator of the unreduced convergents, the denominators of 

 the convergents reduced to their simplest forms be employed, the powers of Z in the 

 constant factor will undergo a diminution. The essential part of this theorem 

 admits of being stated in general terms as follov*rs : — 



" If the quotient of an algebraical function of x by its first dilFerential coefficient 

 be expressed under the form of a continued fraction whose successive partial quo- 

 tients are linear functions of ar, any one of these quotients may be found (to a con- 

 stant factor pres) by taking the sum of the products formed' by multiplying each 

 factor (« — h) of the given function by the square of what the denominator of the 

 immediately antecedent convergent fraction becomes after substituting in it for x the 

 root corresponding to such factor." 



P.S. Since the above was read before the British Association, the theory has been 

 extended by the author to comprise the general case of the expansion of any two 

 algebraical functions under the form of a continued fraction, and has been incor- 

 porated into the paper in the Philosophical Transactions above referred to. 



Light, Heat, Electricity, Magnetism. 



On the Production of Crystalline Structure in Crystallized Powders by 

 Compression and Traction. By Sir David Brewster, K.H., D.C.L., 

 F.R.S., Sf V.P.R.S. Edinb. 



The author had found that by pressing certain crystalline powders against slips of 

 glass, sometimes smooth, sometimes roughened by grinding, with the clean broad 

 blade of a knife or spatula, and drawing it along, he could give to the mass of 

 powder thus treated the same polarizing action on light possessed by large crystals 

 of the same kind ; and which could be given to annealed glass and other non- 

 crystalline substances by mechanical compression, but which they lost when 

 relievid from the compressing force. The author then gave an enumeration of the 

 crystalline powders in which he had succeeded by this compression and traction in 

 producing this polarizing structure, — distinguishing those in which the glass over 

 which they were so distributed required to be rough, from those in which it might 

 be used smooth. He also enumerated the powders which he had tried, but in which, 

 he had not succeeded in producing the same effect. 



On the Optical Phcenomena and Crystallization of Tourmaline, Titanium, 

 and Quartz within Mica, Amethyst, and Topaz. By Sir D. Brewster, 

 K.H., D.C.L., F.R.S., ^ V.P.R.S. Edinb. 



The author, after stating that crystals of titanium within quartz had been long 

 known and attended to, drew attention to the fact that regular crystals of tourmaline, 

 titanium and quartz had been discovered by him within mica, amethyst and topaz } 

 that in some instances these crystals had been found grouped in very regular 

 figures, and that the groups of crystals were sometimes distributed over what were 



I* 



