Exhibit 10.4


RIDER I


Rider to the lease dated 1 July 2004, entered into between Align
Technology B.V. and Stichting Philips Pensioenfonds, currently
TT Amsterdam Project Company Ltd, concerning office space
comprising approx. 1.066 square meters, locally known as the ground floor and
first floor of the building Telepsy at Arlandaweg 161, (1043 HR) Amsterdam the
Netherlands (the “Existing  Lease”).


This Rider is hereby executed by and between:


Align Technology B.V., established in Amsterdam (1043 HR) at Arlandaweg
161,  in this matter duly represented by Mr T. Prescott and Mr. E. Bullington,


listed in the Commercial Register in Amsterdam under number 34207531


hereinafter referred to as the “Lessee”,


and


TT Amsterdam
Project Company Ltd. c/o Padget-Brown Trust Company Ltd, established in West Wind Building,
Harbour Drive, P.O. Box 1111, at George Town, Grand Cayman, British West Indies


in this matter duly represented by Mr H.D. Garrison,


listed in the Commercial Register under number PB-168944


hereinafter referred to as the “Lessor”


WHEREAS:


·                  The Lessee and the Lessor have agreed on the extension
of the leased premises to include part of the second floor of the building of
Telepsy, 3 parking spaces and a new lease term for a period of 5 years with a
renewable 5 year option.


·                  The Lessee and the Lessor wish to lay down
the agreements made regarding this extension and new term in writing.


THE PARTIES HAVE AGREED TO THE FOLLOWING:


Extension accommodation and new term


1.               The lease term of the Existing Lease is hereby
amended to include a 5 year term commencing on 1 June 2007 (the “ New Lease Term Commencement Date”) with an option to renew
for an additional term of 5 years.


2.               On the New Lease Term Commencement Date,  the leased premises referred to in Clause 3.1
of the Existing Lease shall be amended to include a part of the second floor
and 3 additional parking spaces such that the total leased premises shall
comprise approximately 1.526 m2 and 13
parking spaces.


	
  Initials of the Lessor 

  	
  Initials of the Lessee

  



 













Conditions


Lease
notice period:


The notice period
referred to in Clause 3.3 of the Existing Lease shall be amended to include a
12 (twelve) month notice period.


The total
leased property:


The total leased property
shall comprise approximately 1.526 m2 of
office space located on (i) the ground floor (487m2), (ii) on the first floor (579 m2) and (iii) part of the second
floor (460m2) according
to the NEN 2580, as well as 13 parking spaces.


The Rent
amount provided Pursuant to Clause 4.1 of the Existing Lease shall be


amended as follows:


	
  Current office space:

  	
   

  	
  € 204,- excl. of VAT

  
	
  (current rent)

  	
   

  	
   

  
	
   

  	
   

  	
   

  
	
  Extension office
  space (460 m2):

  	
   

  	
  € 180,- excl. of VAT

  
	
   

  	
   

  	
   

  
	
   

  	
   

  	
   

  
	
  Parking spaces:

  	
   

  	
  € 1.514,- excl. of VAT

  
	
  (current rent)

  	
   

  	
   

  



 


Creditable
advance on service charges:


EUR 22 per m2, excl. of VAT. The service
charges are invoiced quarterly in advance. Amounts paid are set off annually
based on the costs actually incurred.


Rent-free
period


Tenant will receive a rent-free
period (excl. service charges) for the leased property of the extension office
spaces (460 m2,
second floor) and the 3 parking spaces for 1 1⁄2 years (from 1 june 2007 until 1 December
2008).


State of delivery


The office space
will be delivered in its current condition, with new carpet.


Right of first refusal

The lessee has a right of first refusal on the remaining
office space on the 2nd, 3rd and 4th floors. If another party shows an
interest in the remaining office space, the lessee will be notified in the form
of a written offer (sent by registered post). Within 5 working days of receipt
of this notification, the lessee will indicate by means of a letter sent by
registered post whether it accepts the offer. If the lessee does not wish to
accept the offer, or does not respond before the set deadline, the lessor will
be free to let the space to third parties subject to conditions that the lessor
itself will specify in so far as not determined otherwise in this agreement.


Remaining conditions present lease contract


All the remaining
conditions of the Existing Lease contract which has not been laid down in this
proposal don’t change and will be remain effective.


	
  Initials of the Lessor 

  	
  Initials of the Lessee

  



 













Acceptance


Upon signature of
the lease. The remaining time until the New Lease Term Commencement Date may be
used to furnish the extension of the leased property.


Miscellaneous


Unless otherwise agreed in this rider, the
provisions of the Existing Lease shall remain in full force and effect.


Signed,
sealed and delivered in three copies.


	
  August 3, 2007

  	
  August 3, 2007

  
	
   

  	
   

  
	
   

  	
   

  
	
  p.p. the Lessor:

  	
  p.p. the Lessee:

  
	
   

  	
   

  
	
   

  	
   

  
	
  (signature)

  	
  (signature)

  
	
  /s/ Jeffrey
  D. Haroldson

  	
   

  	
  /s/ T. Prescott

  	
   

  
	
   

  	
   

  	
   

  	
   

  
	
   

  	
   

  	
  (signature)

  	
   

  
	
   

  	
   

  	
  /s/ Eldon M. Bullington