24 



SCIENCE. 



[Vol. V., No. 101. 



expressions for the various constants (in terms of 

 those of the two-wire system), when the same plant 

 of lamps is divided up (with n conductors, each of 

 whose cross-sections is k times that of one of the 

 first) into n — 1 equally-balanced circuits, with n — 1 

 dynamos in series, as in the Edison system figured 

 on p. 477, vol. iv. (the same energy being developed 

 in the lamps as at first), will then be : — 



(n — l)' 2 . r — the resistance of the lamps. 



R 



it == the resistance of the two outside conductors. 



— the current in the same. 



n - 1 

 CR 



i y> i. + Q r = En-i — difference of potential 

 > ■ ' at terminals of each of the ?i — 1 



dynamos. 



C 2 R 



— — = heat-energy wasted in two outside con- 

 In 1) . k ductors. 



(n — l) 2 . r = C z r = energy developed in 

 the lamps. 



(n - l)* 



n n 



x- . k . C 2 R = ^ . k = interest on cost of n con- 

 ductors. 



The energy consumed in the lamps (C-V) is the 

 same as at first, as shown by Professor Carhart, and, 

 being constant for the plant of lamps, need not be 

 further considered. The total running-expense, then, 

 due to conductors (including interest on their cost), is 



. k 



(n - I)"- 



in terms of C 2 R. This should be a minimum. Its 

 first differential coefficient with reference to k, placed 

 equal to zero, gives 



1 |2 

 k ~ n- l\in 



as the most economical section. This gives, for the 

 minimum value of the total running-expense, 



w-lS|2n-lS|2~n-l V Zn ' 



the interest and heating-cost being equal, as they 

 should. The same value of k gives 



1 l» 



E n -1 = 7 Jtj. CR + 



n — 1 \2 



Or 



as the corresponding difference of potential at the 

 terminals of each of the n — 1 dynamos. 



Substituting now in these different expressions, 

 and also in corresponding ones for the Edison sys- 

 tem and for Professor Carhart's plan, values of n 

 from 2 to 6, we have the various data given in the 

 following table. 



OD 

























o 

























3 



Cross- 



Difference of 



Interest 



Heat- 



Running- 



T3 



section 



potential at ter- 



waste in 



expense 



O 



of 



minals of 



on con- 



con- 



of con- 



O 



each. 



each dynamo. 



ductors. 



ductors. 



ductors. 



i 













Edison's system. 





1 



(?i-l) 2 



(71— 1) (»— 1) 



n 



(72 - l) 2 



n 

 2(7 i -l)2 +1 





(n-\y- 



2(71-1)2 



(n - l)* 



2 



1.000 



1.000 CR + Cr 



1.000 



1.000 



2.000 



3 



0.250 



1.000 " 



0.375 



1.000 



1.375 



4 



0.111 



1.000 " 



0.222 



1.000 



1.222 



5 



0.062 



1.000 " 



0.156 



1.000 



1.156 



6 



0.040 



1.000 " 



0.120 



1.000 



1.120 



Professor Carhart's plan. 



1 



1 



1 

 r . CR + Cr. 



n — 1 



n 



1 



n + 1 



"! 



n-1 



2 (n - 1) 



71- 1 



2(71-1) 



2 ! 



1.000 



1.000 CR + Cr 



1.000 



1.000 



2.000 



3 1 



0.500 



0.500 " 



0.750 



0.500 



1.250 



4 



0.333 



0.333 " 



0.667 



0.333 



1.000 



5 



0.250 



0.250 " 



0.625 



0.250 



0.875 



6 



0.200 



0.200 " 



0.600 



0.200 



0.800 



The most economical plan. 



11. 



A.JI 



11— \ " ft 



n-1 * 2 



Tl-l T 2 



JL.i/'i 

 7i-i r 2 



n-1 



2 

 3 

 4 

 5 



6 



1.000 

 0.408 

 0.236 

 0.158 

 0.115 



1.000 CR + Cr 

 0.612 " 

 0.471 " 

 0.395 " 

 0.346 " 



1.000 

 0.612 

 0.471 

 0.395 

 0.346 



1.000 

 0.612 

 0.471 

 0.395 

 0.346 



2.000 

 1.225 

 0.943 

 0.791 

 0.693 



Some very interesting comparisons of relative ad- 

 vantages might be noted, did space permit. The most 

 important is the rapid increase in the ratio of heat- 

 energy to capacity of conductor in the Edison sys- 

 tem, which might make it necessary to lay the wires 

 so as to admit of pretty free radiation of heat. 



This question of temperature of electric-lighting 

 conductors promises to protrude itself the more they 

 are laid underground. The desideratum for an insu- 

 lating covering would seem to be a non-conductor of 

 electricity and good conductor of heat, — apparently 

 inconsistent qualities. Perhaps the eventual solution 

 will be in bare or loosely covered wires on highly 

 insulated points of support, thus admitting of free 

 radiation of heat, like aerial lines. H. M. Paul.. 



Washington, Dec. 15. 



Mr 



Sun-spots. 



Todd, in a recent number of Science, speaks 



17X1. XUUUj ILL a, ICtClll 11UU1UU Ul (U'UC'OIIOO, C3^J\^ai\J 



of Sept. 23, 1883, as the last day of that year on which 



