HYDROSTATICS. 



Roots of the same letter or quantity may be t/t 

 tit, ii- j'i iiftimiiil. erpoi 



I othw fractions, must be reduced 

 ( , u ooaunoo ;. -nominator before they can be united 



1 i 1,1 | 4. | I 



MPLE. Thus a* X a 8 = a* "* " =* a* * = a'. 

 values of the roots are not altered by reducing their 

 indices to a common denominator. 



fore the first faoto> a* = a" 

 An>l tho second a* = a* 



Hut J = a* X a* X a^; and a' = a* X a*. 

 .o product therefore is a* x a* x a* x a* X a" = a*. 



N I' [nallii t ancos of this nature, the common denominator 

 iuilicfs denotes a certain root; and the sum of the 

 shows how often this is to be repeated as a factor 

 to produce tho required product. 



, i m n m + n 



EXAMPLE Thus a" x a m = a 1 "" x a mn = a '"" . 

 Any quantities may bo reduced to tho form of radicals, and 

 may then bo subjected to tho same modes of operation. 



t n+j . 



Thus y 3 x y = y s + " = y ; and or x o>" = x n 

 N.B. Tlie product will become rational whenever the nume- 

 rator of tho index can be exactly divided by the denominator. 



EXAMPLE. Thus a 3 x a$ X a* = a' = a. 

 When radical quantities which aro reduced to the same index 

 have rational coefficients, tho rational parts may bo multiplied 

 T, and their product prefixed to the product of the 

 radical parts. 



EXAMPLE. Multiply a \/b into c Vd. 



The product of tho rational parts is ac. 

 The product of tho radical parts is Vbd. 

 And the whole product = ac \/6d. Ans. 

 But in cases of this nature we may save the trouble of re- 

 ducing to a common index, by multiplying. 



-Thus ax* into ld*= ax- bd*. 

 EXERCISE 53. 



EXAMPLE.- 



Ans. 



1. Multiply */a + m into V "< 



2. "Multiply Vdx into -/.hy. 



3. Multiply a* into i^. 



4. Multiply ( a+ y)~ into (b + ?i)~ 



5. Multiply o m into x*. 



6. Multiply J&cb into V2J*7 



7. Multiply (a*!/ 1 )* into (a*y) . 



8. Multiply 3y* into y*. 



9. Multiply (a + b)* into (a + b)*. 



i i 



10. Multiply (a - y)" into (a-y)m. 



13. Multiply Kn into afq. 



14. Multiply oft m t or*. 



15. Multiply a 2 into a*. 



16. Multiply (a + b)*into (a + b)*. 



17. Multiply a* into a*. 



18. Multiply ax* into 6df 



19. Multiply o(b+r)* into y(bz)*. 



20. Multiply a Vy s into b >//iy. 



21. Multiply a Jx into b V*. 



22. Multiply tut~^ into by 



~^ 



23. Multiply x V3 into y ^9. 



24. Multiply * \/ob into ' \/ob* 



11. Multiply x* into *>. 



12. Multiply y* into y*. 



If the rational quantities, instead of being coefficients to the 

 radical quantities, are connected with them by the signs -f- and 

 - , each term in the multiplier must be multiplied into each 

 term of the multiplicand. 



EXAMPLE. Multiply a +>Sb 



Into c + */d 



ac -\-c/b 



a \/d + */bd 



Hence we deduce the following 



GENERAL RULE FOR MULTIPLYING RADICALS. 



Radicals of tlie same root are multiplied by adding their 

 fractional exponents. 



If the quantities have tlie same radical sign or index, multiply 

 them together as you multiply rational quantities, place the pro- 

 duct under the common radical sign, and to this prefix the 

 product of their coefficients. 



If the radical* curt compound qmmHHsi, eatk fen* in m 



u-r must be multiplied into 

 fcy writing the terms out <*/frr another, ' 

 tht sign of multiplication between them. 



. Mallipl, ( - , .. V-) t*. 



10. Multiply 2S ( <f e) * Wo 

 -8( + cr. 



12. Multiply* ViBlosM*. 



L Multiply IS v/* into 10 ' V. 



2. Multiply 4 ^iatof^**-*. 



3. Multiply aji into w'^y. 



4. Multiply JV) into iV|. 

 A. Multiply S** into 4<i. 



8. Multiply 1- 



KEY TO EXEKCI8E8 IH LK88OH8 W ALOKBRA.-UIX. 

 EZKRCIU 48. 



4. (27 (a - ))* or 



2. 46 V<-. 



4. 



5. a(a - b)' 



6. 3a(2b)*. 

 7. 



8. 



a (a)"uidt. 



()"* (/)-' 



10. (2 ) 1 and (,') k . 



11. V(a + b)" and 



9. (16ab*)'. 



10 f-J^^J\^ 

 \ a'b* + t* / 



11. V8. 



12. vieb^e: 



13. V150. 



14. V| 



15. (5)* and (6)*. 

 EXERCISE 50. 



14. 



14. 



U. (;i mad (*)*. 



17. 



2. 3^o. 

 3. 



1. 7 -/I 



2. 



3. 



4. 9V5. 



1. -2v/ay. 



3. 8h*. 



4. (a- 



4. 

 5. 



6. 6^b. 

 EXERCISE 51. 



5. 32V2. 



6. 191^3. 



7. 16 Vb. 



EXERCISE 52. 

 5. a c 

 7. (b-v)Vty. 



16. () awl (*jl. 



17. 7,'J. 



18. 9v^3. 



19. V^_ 



20. 28V&7 

 21. 



23. 



9. llx 

 10. 7a> 



9, 



10. 2VS 

 1L ^5. 

 12. 4a- 



HYDROSTATICS. IX. 



HTDRATTLIC BAM MACHINES FOB RAISING WAI 

 WHXKL. 



A VERT ingenious and useful piece of apparatus, invented by a 

 celebrated Frenchman named Montgolfier, must be noticed, as 

 not only is it very useful, but it involves several of tbe principles 

 we have already considered. It is frequently required to raise 

 a quantity of water to some elevation, and this machine, which 

 is called the Hydraulic Bam (Fig. 30), accomplishes this by the 

 momentum acquired by the fall of a current of water. As a 

 considerable fall of water is desirable, and it is likewise impor- 

 tant that it should remain nearly constant, a dam is usually 

 bn : lt across the stream, so as to form a reservoir, which ovei 

 flows when full, and thus maintains a uniform level. From this 

 reservoir pipes are brought along tbe bed of the stream to join 

 on to D. The other end of this pipe is closed. A ralT* opening 

 downwards is, however, inserted at c, near the ad.jne spindle 

 of this passes through a guide, so as to keep it TertiosJ, and it 



