## Sentence Examples

**Thus****to****obtain****Stroh'****s****theory****of****seminvariants****put**b1=0**-**1+a2+ï¿½ï¿½.+0**-m****These****seminvariants****are****said****to****form****an****asyzygetic****system**.- -
**zn**+9 1 -z2.1 -z3....1**-**z8;**and****since****this****expression****is****unaltered****by****the****interchange****of****n****and****B****we****prove****Hermite'****s****Law****of****Reciprocity**,**which****states****that****the****asyzygetic****forms****of****degree**0**for****the**/**t****ie****are****equinumerous****with****those****of****degree****n****for****the****The****degree****of****the****covariant****in****the****variables****is****e**=**nO-**2w;**consequently****we****are****only****concerned****with****positive****terms****in****the****developments****and**(**w**, 0,**n**) - (**w**-**r**; 0,**n**)**will****be****negative****unless****nO****It****is****convenient****to****enumerate****the****seminvariants****of****degree**0**and****order****e**=n0**-**2w**by****a****generating****function**;**so**,**in****the****first****written****generating****function****for****seminvariants**,**write**z2**for****z****and****az****n****for****a**;. **Putting****n****equal****to****co**,**in****a****generating****function****obtained****above**,**we****find****that****the****function**,**which****enumerates****the****asyzvgetic****seminvariants****of****degree**0,**is**1 1**-**z2.1**-**z3.1**-**z4....1**-**z0**that****is****to****say**,**of****the****weight****w**,**we****have****one****form****corresponding****to****each****non-unitary****partition****of****w****into****the****parts**2, 3, 4,...0.**Now****the****symbolic****expression****of****the****seminvariant****can****be****expanded****by****the****binomial****theorem****so****as****to****be****exhibited****as****a****sum****of****products****of****seminvariants**,**of****lower****degrees****if****alai**0**-**2a2 +...+crea0**can****be****broken****up****into****any****two****portions**(**alai**-1**-**0**-**2a2**-**1-ï¿½ï¿½ï¿½ +**asas**) +(**as**+1as +1 +**o-**8+2as+2+ï¿½ï¿½ï¿½ +**ooae**),**such****that**Q1 +a2+...**Solving****the****equation****by****the****Ordinary****Theory****Of****Linear****Partial****Differential****Equations**,**We****Obtain****P****Q**1**Independent****Solutions**,**Of****Which****P****Appertain****To**S2Au = 0,**Q****To**12**B****U**=0;**The****Remaining****One****Is****Ab**=**Aobl****A**1**Bo**,**The****Leading****Coefficient****Of****The****Jacobian****Of****The****Two****Forms**.**This****Constitutes****An****Algebraically****Complete****System**,**And**,**In****Terms****Of****Its****Members**,**All****Seminvariants****Can****Be****Rationally****Expressed**.**A****Similar****Theorem****Holds****In****The****Case****Of****Any****Number****Of****Binary****Forms**,**The****Mixed****Seminvariants****Being****Derived****From****The****Jacobians****Of****The****Several****Pairs****Of****Forms**.**If****The****Seminvariant****Be****Of****Degree**0, 0'**In****The****Coefficients**,**The****Forms****Of****Orders****P**,**Q****Respectively**,**And****The****Weight****W**,**The****Degree****Of****The****Covariant****In****The****Variables****Will****Be**P0**Qo**' 2W =**E**,**An****Easy****Generalization****Of****The****Theorem****Connected****With****A****Single****Form**.**The****Number****Of****Linearly****Independent****Seminvariants****Of****The****Given****Type****Will****Then****Be****Denoted****By**(**W**; 0,**P**; 0',**Q**) (**W**; 0,**P**; 0',**Q**);**And****Will****Be****Given****By****The****Coefficient****Of****A****E****B****E**'**Z****W****In****L****Z**1**A**.**For****Two****Forms****The****Seminvariants****Of****Degrees****I**,**I****Are****Enumerated****By**1**Z**,**And****The****Only****One****Which****Is****Reducible****Is****Ao**0**Of****Weight****Zero**; 1**Hence****The****Perpetuants****Of****Degrees****I**,**I****Are****Enumerated****By**11 1 ï¿½**Z**1Zz'**And****The****Series****Is****Evidently****A****O****B**1**Aibo**,**A**0**B**2**A****B**A2Bo,**A****O****B**3**A****L****B**2**A**2**B**1 A3Bo,**One****For****Each****Of****The****Weights****I**, 2, 3,..**Ad****Infin**.- 1
**Ze****An****Expression****Which****Also****Enumerates****The****Asyzygetic****Seminvariants**,**We****May****Regard****The****Form**,**Written**,**As****Denoting****The****General****Form****Of****Asyzygetic****Seminvariant**;**A****Very****Important****Conclusion**. **Thus****what****have****been****called****seminvariants****are****not****all****of****them****invariants****for****the****general****substitution**,**but****are****invariants****for****the****particular****substitution****xl**= X11 +**J-**s12,**X**2 = 112**Again**,**in****plane****geometry**,**the****most****general****equations****of****substitution****which****change****from****old****axes****inclined****at****w****to****new****axes****inclined****at****w**' =13 -**a**,**and****inclined****at****angles****a**, l3**to****the****old****axis****of****x**,**without****change****of****origin**,**are****x-sin**(**wa**)**X**+**sin**(**w**-/3)**Y****sin****w****sin**' _sin**ax****y****sin****w****a****transformation****of****modulus****sin****w**'**sin****w**'**The****theory****of****invariants****originated****in****the****discussion**,**by****George****Boole**,**of****this****system****so****important****in****geometry**.