190 The Rev. F, H. Hummel on 



a number of open currents, which render the gas incandescent 

 in their path, and reach so much the further the greater the 

 exhaustion is. If, when the exhaustion is not very great, the 

 length of the discharge issuing from the kathode is less than 

 the interval between the kathode and the next point of dis- 

 charge (from which the first positive layer issues), there then 

 must exist between the kathode-light and the first positive 

 layer a space traversed by no current, in which therefore there 

 is no luminosity, the so-called " dark space." 



If the current-length of the kathode-discharge increases in 

 consequence of increased exhaustion, so that it is equal to the 

 interval between the kathode and the next point of discharge, 

 then the kathode-rays reach the positive light and the dark 

 space vanishes. If the length of the kathode-current become 

 still greater than that interval, then the kathode-light advances 

 into the space into which also flows a current from the second 

 point of discharge, and the kathode -light penetrates into the 

 positive light. In exactly the same way is explained the pro- 

 duction of the dark space between each bundle of secondary 

 negative light and the layer which follows it, and the dark 

 spaces which the layers show between them at comparatively 

 small exhaustion, whilst they are in contact when the ex- 

 haustion is greater. 



In the same way the phenomena not to be explained by 

 previous views, those obtained (pp. 178-183) with kathodes of 

 different forms and positions, contain nothing more that is 

 puzzling ; and there is no more need to suppose a to-and-fro 

 motion of electricity, or of a repeated zigzag path, nor of a 

 new and non-luminous mode of discharge. 



[To be continued.] 



XXVIII. Evolution by Subtraction. 

 By the Eev. F. H. Hummel, M.A* 



" \\7"HEN we require to find a root (say, e. g., the square 

 ▼ V root) of an arithmetical quantity, we begin by ' point- 

 ing,' i. e. dividing off the digits of our given quantity into equal 

 groups, starting from the units' place, each group containing 

 a number of digits equal to the index of the required root; the 

 last group on the left containing so many digits as happen to 

 remain to it. On this last group we commence operations, 

 finding the square root of the square number next below it in 

 numerical value. The remaining steps are simply successive 

 solutions of a continually recurring equation of the form 

 n 2 =a 2 + 2aa+a 2 } in which, by approximating to x at each 

 step, and then adding its approximate value to a for the next 

 * Communicated by the Author. 



