424 
Direct cosecants of Prcke dnciinstions te the cnlleete PR: CTS- 
Methed- Gonomerny) ; but as the 
—\~" inclination of the line P’ 
Fig. 17. 
Fig. 15. 
Poel 
and at last the two lines coincide 
of the ratio of PH PH ma matio of equality, rar oa 
the chord and are are of an intermediate be- 
tween PH and P’H’, the limit of their ratio must also 
be that of equality. 
75. Genes i A to hea arigih of he. xestenguie: 
in 
then, because the arc PP’, and the line PK, are corre- 
sponding increments of = and 2, the expression for the 
fluxional ratio $= will be equal to the limit to which 
the fraction "1?" approaches, when QQ’=PK, the 
increment of x, is diminished indefinitely, (art. 23.) 
Now we have seen, that the arc PP” is of an intermediate 
magnitude between the lines PH, P’H’, which touch 
the curve at P and P’; therefore the pai 
fastege of an intermediate magnitude between these 
PH’ 
But we have seen, (art. 74.) that 
two, => PK, and Se 
when P’ to P, the ratio of P’H’ to PH ap- 
proaches to» ratio of equality ; therefore the limit of 
ea will ey! and eonsequently the limit of 
PK’ 
a spp’ PH a? = PH 
PK ill also be —.— 7K" Hence we have =~ = 7K" 
And since, by trigonometry, Fe = secant of the angle 
HPK=,/(1-+tan2 HPK), a by art. 67, formula 
(1), tan, HPK = $2 therefore 72 = (1455), 
and hence again 
de=y/(ds*4dy?) (1.) 
From which that the square of the 
fosion of the ar, isthe sumo) the aguarer of Ufone 
of the rectangular co-ordinates. 
76. In the case of curves expressed | a 
tion, (Fig. 15,) patente rey M4 he bp ng which 
the radius AP=r revolves put» forthe variable angle 
which r makes with AB, a given by position pas- 
sing A; and the Gite tobe aude re- 
ferred to this line as an axis rectangular co-ordinates, 
Fy origin at A, so that AQ=« and QP=y, 
z=rcos.v, y=r sin. v, 
pce eer gh aig bed 
=—ydu+ <dr, 
dy=r cos, vdv+-sin,vdr, 
eran 
dtaydty Fant ~<! dedy, 
dY=z'd vt aLangtt2 dudr; 
Therefore, . 
detdy=(st4+y)(dt42P), 
but 44y'=r’, and dz*4dy'=d 2, (art, 75.) 
FLUXIONS. 
ves a the PE, 
ms erent enna mens 
presently shew), we can find also the of the 
more points in the curve as we please. 
Take another point P’ in the curve, and draw a tan- 
gent P’E’, meeti a ppd in G; also draw AE’ 
besitos rai rpendgcular AE by py and 
Wy ? Pee 
be regarded as functions of u. Put h for 
pear te a for an ero 
new values t, correspo to u+ new 
value mid evn the disposition of the lines in the fi. 
, we have 
E'P/—EP=E’G+ GP’—(EH4+ HG—G 
= PG j{GP'_RH—tHG_—GR); 
and dividing by the are /, 
es fhe tre 
a HG—GE’ 
h 
trigonometry, EH tan. h; and, because 
AEH GEA are emilar, EG= GH x 
Nom 
seg ‘ore, by substituting, we get 
de — PG+GP’ tan. 1—cos.h 
ne ee ee 
now, h to be diminished indefinitely, 
Lome P’ will approach to P; we have then, ‘bait 
—=t=4 (art, 23), and bait a (art. 
23. and 73.) ; and because limit tant <1 (art. 78.) 5 
hp AE Wasnt dai du. 
er bo the Bie penton sora sey a4 
arc , : same 
Sate fen Os pee 
2 du 
To express ¢ by meat of pai u, we have 
‘ 
OE 
nh ae 
