Aug. 15, 1884.] 



♦ KNOWLEDGE ♦ 



145 



form green, and not white at all. 3. Flints are ajrirrepations of 

 silica round shells, sponges, cchinites, &e., which died and sank to 

 the bottom of the ocean which deposited the chalk. There were 

 thermal springs then as now, and you know that hot water will 

 <lissolve more or less silica. 4. Ciel et Ten-e is published in 

 Brussels on the 1st and 15th of each month in the pamphlet 

 form. The subscription to it is 10 francs (i.e., 83. 4d.) a year, for 

 which it is sent free by post. — Peecv Gkeg has seen a table move 

 " without human or mechanical agency." Scores of people heard 

 a musical-box play, and instantly cease playing, at the word of 

 command, in the presence of a vagabond called Jlonck ; the box 

 standing absolutely isolated and in full view in the middle of a 

 table without any cloth on it, and Jlonok's hands being held by the 

 persons sitting on each side of him. When this highly respectable 

 "medium" subsequently found himself in a police court, the modus 

 nperandi of this quasi-miraculous phenomenon was exposed at once. 

 If we were to attribute everything for what we wholly fail to find 

 an explanation to the action of " disembodied intelligences," then 

 surely must we credit Bautier, Frikell, Hermann, and ilaskelyne 

 and Cooke with very intimate communion indeed with another 

 world. — J. Greevz Fisher sends me a letter pointing out — in effect, 

 albeit, in slightly more euphemistic terms — how very foolish I am 

 not to reform my spelling straightway. He accompanies this by 

 some little tracts which I will read. When I am convinced by 

 them that " Xolij " is a better form of writing the title of this 

 journal than Knowledge, I will consult the Messrs. Wymau as to 

 the policy of substituting the first form of heading for the existing 

 one. — H. WoRMAlD. A "singing mouse," poor little beast, has an 

 inflammatory affection of its air passages. — G. A. Spottiswoode. 

 This journal addresses students of every branch of science, and the 

 article to which you take exception was one possessing great 

 physiological interest. Knowledge iu no sense enters into com- 

 petition with such publications as Little Folif or Aunt Judy's 

 Magazine. — Algernon Br.\v. Will you forward your address to 

 the oflice of this paper, as a letter is lying here for yon ? — J. Elgie. 

 (1) How can yon expect a body shining, ex hypothesi, only by 

 reflected light, and never departing by many seconds of arc from 

 its primary, to be visible with any power, high or low, at the 

 stupendous distance of Algol ? (2) D' Arrest, in 1805, found a 

 10-llth Mag. Star somewhere near the place in Cassiopeia. The 

 new star blazed out in Nov., 1572, and this was afterwards seen by 

 Espin in 1S7S. (3) You can divide Castor with a power of 80 on a 

 2-inch object-glass. ( t) I cannot say offhand how many physically 

 double stars are now known, but there are certainh- some hundreds. 



0av iHatftcmatiral Column. 



EASY LESSONS IN CO-OEDINATE GEOIIETEY. 



By Richard A. Peoctob. 



63. Prop. — To determine the equation to a straight line parallel to a 

 ^iven straight line. 



Let the equation to the given straight line be 



y = mx + c (i) 



then since a striiight line parallel to the given straight Hne must be 

 inclined to the axis of x at the same angle, the equation required is 



y='m.r + c (ii) 



an equation differing only iu the constant term from equation (i). 

 It follows that the equation — 



At + B w + C=0 

 represents a straight line parallel to the line whose equation is 

 A,i-i-B!/-l-C=0 

 The equation to a straight line parallel to the straight line repre- 

 sented bv the equation 



- + 1=1 

 a 



tnay be shown from the same consideration to be 



-^\ = 1 



ra rb 



a relation which may readily be established independently ; since 

 it is obvious that the intercepts of two parallel lines on the axes 

 of r and y must be respectively proportional. 



64. Prop. — To determine the co-ordinates of the point of intersection 

 of two straight lines u-hose equations are given. 



Let the equation to the two given straight lines be 

 y = mi.c-t-c, (i) and 'j = m.2 x + c, (ii) 

 Solving these simultaneous equations, we obtain 



Ci — c-y Cim,i — r.>mi 



x = ~ y = = ' 



m^ — wij ■ /*io — 7>ij 



Since both equations are satisfied when we give to •: and 1/ the above 



values, it follows that the point which has these coordinates is a 

 point on both lines, — that is, is the point in which the two linos 

 intersect. 



If 7?ij = m, the values above obtained for .c and y are indefinitely 

 great. This is the analytical expression of the relation that parallel 

 lines never meet. 



If the equations of the lines be given in the form 



AiJ +B, ly 4-Ci = (i) and A^i-i-Bjy-HC- = (ii) 

 we obtain 



B,C.,-B„Ci A,Co-A„C, 



" = A,B,-A,Bi' """^ y = A,B,-A.B., 

 65. Prop. — To determine the condition that three straight lines 

 ichose equations are given may pass through a point. 

 Let the equations of the three lines be 



!/ = mja;-pC] (i), y = nuj> + c.2 (ii), and y = ni3T-i-i-3 (iii). 

 Then the x co-ordinate of the point of intersection of (i) and (ii) 



' ^ and the a;-co-ordinate of the point of intersection of (i) 



iHj — nil, 

 and (iii) is — ! — 



Hence, if (i) (ii) and (iii) meet at the same 



point we must clearly have 



Ci — C.1 



Ci-Cj 



(0 



(ii) 



tiv; — nti 7713— mi 

 that is 



<^i™3 ~ '^a'"! + '^znu — c^m-^ + c^m^ — Ciiii.. =0 (iv) 



In fact (iv) is the algebraical condition that the three equations 

 (i) (ii) and (iii) may be equivalent to only two independent 

 equations. 



We know from algebra that if the three equations (i) (ii) (iii) 

 are equivalent to only two independent equations, that is if con- 

 dition (iv) hold, then when (i) (ii) and (iii) are multiplied respec- 

 tively by the quantities c^m^ — c^m.,, C3"»i — Ciwij, and c,n!.;— c.,m,, and 

 added together, both sides of the resulting equation vanish iden- 

 tically. Hence we obtain the following rule, which is frequently 

 useful in practice. If three equations representing straight lines 

 can be multiplied by such quantities that the sum of the resulting 

 equations expresses identical equality, the straight lines represented 

 by the three equations all pass through a single point. 



66. Prop. — To determine the form of the equation to a straight 

 line passing through the intersection of two given straight lines. 



If these equations to the two straight lines are 

 A,! +Biy-HCi = 

 A,vr + B.,y-HC2=0 



We may proceed to a full-length solution as follows : — 

 Solving (i) and (ii) we get for the point of intersection 

 BiC.-B.,C, ^ A.,C,-A,C., 



^=aib:-a:b, ^"'^ ^==a;b,-a,b; 



.'. by the equation to a straight line passing through the point of 

 intersection of (i) and (ii) is of the form 



L a,b.-a.BiJ L AiB»-a.,bJ *■ ' 



or (AiB2-A.,Bi)(Ai-hBt/)-f(BB.,-AA.,)C,-i-(AA,-BB,) 



C;=0 (iv) 



Now if we multiply (i) by (B B.-A Ao) and (ii) by (A Ai-BBJ 

 and add, we get for the co-eflicient of x 



A,B B,,-A A,A.-(- A AiA;-A;B Bi = B(AiB;-A.,Bi) 

 and for the co-eflicient of v 



BB,B.-AA,B, -hAAiB.,-BBiB.: = A(AiB.,-A;B,) 

 or for the resulting equation the form (iv). 

 We might simply have proceeded thus 



A,j! + B,!/-iCi = (1) 



A»3;4Bov-hCo = (2) 



any equation of the form 



Z(A,.'' -I- Bit/ -1- Ci) + m(A.;7- -I- B;!/ + C-) =0 (3) 



which is also obvious since whatever values of t and y satisfy (1) 

 and (2) must also satisfy (3). It is also obvious that every equation 

 to a line passing through the intersection of (1) and (2) can be 

 put in the form 3, seeing that the ratio B : A of the preceding 

 article is the tangent of the angle made by a straight line through 

 (i) (ii) with the axis of x, and as we have seen to give B and A 

 Bnitable values, all we require is that the following equations shall 

 hold. 



i = EB;-AA, 

 and »i = AAi — BB, 



The method of the preceding article leads directly to the very 

 valuable method of treating certain problems on the straight line 

 called the Method of Abridged Notation. I do not here dwell upon 

 it because any one who is likely to require much information on the 

 subject would go to treatises where the methods of co-ordinate 

 geometry are more fully dealt with than they can be here. 



